# Tuning magnetic interactions of Co and 4$d$ transition-metal atomic bilayers
on Re(0001) via interface engineering

Souvik Paul<EMAIL_ADDRESS>School of Physics,
Indian Institute of Science Education and Research Thiruvananthapuram,
Thiruvananthapuram, Kerala 695551, India  Peter Gr$\ddot{u}$nberg Institut and
Institute for Advanced Simulation, Forschungszentrum J$\ddot{u}$lich and JARA,
52425 J$\ddot{u}$lich, Germany.  Stefan Heinze  Institute of Theoretical
Physics and Astrophysics, Christian-Albrechts-Universit$\ddot{a}$t zu Kiel,
Leibnizstrasse 15, 24098 Kiel, Germany  Kiel Nano, Surface, and Interface
Science (KiNSIS), University of Kiel, Germany

(August 28, 2024)

###### Abstract

Employing ab-initio density functional theory (DFT), we performed a systematic
investigation of the electronic structures and the magnetic properties of
atomic bilayers composed of a 4$d$ transition-metal layer (Rh, Pd and Ru) and
a Co layer on the Re(0001) surface. Our DFT calculations reveal the influence
of the bilayer composition and their stacking sequence on the magnetic ground
state and magnetic interactions. We obtain the magnetic interactions by
mapping the total energies onto an effective spin Hamiltonian which contains
the exchange interaction and the Dzyaloshinskii-Moriya interaction (DMI) as
well as the magnetocrystalline anisotropy energy (MAE). We observe noticeable
changes in bilayer hybridization due to variation in bilayer composition and
overlayer symmetry, leading to significant variation in magnetic interactions.
In all considered systems, the nearest-neighbor exchange interaction is
ferromagnetic, however, the value varies by up to a factor of 5. The
effective, nearest-neighbor DMI constant exhibits variation in sign over the
films considered, favoring either right- (clockwise) or left-handed
(anticlockwise) cycloidal spin spirals. The value of the DMI changes by up to
a factor of 27 among the films. For most of the systems, the MAE favors an
out-of-plane easy magnetization axis, however, for hcp-Co/Rh and hcp-Co/Ru
bilayers on Re(0001), it prefers an in-plane magnetization axis. The magnitude
of the MAE varies from a small value of about 0.1 meV/Co atom up to about 2
meV/Co atom for Co/Pd bilayers. The spin spiral energy dispersion curve rises
quite quickly close to the ferromagnetic state for films in which the Co layer
is adjacent to the vacuum indicating a large effective exchange constant which
stabilizes a ferromagnetic ground state in Co/$4d$ bilayers on Re(0001). The
energy dispersion curve becomes flatter for films with a Co layer that is
sandwiched between a $4d$ overlayer and the Re(0001) surface. In this case,
the exchange constant is much reduced and the ground state is determined by
the competition among the exchange interaction, favoring the FM state, the
DMI, which favors cycloidal spin spirals, and the MAE, which disfavors spin
spirals over the FM state. As a result, hcp-Rh/Co/Re(0001) shows a spin spiral
ground state driven by DMI with a period of 13 nm while the other films
exhibit a ferromagnetic ground state. The spin spiral energy dispersion of
hcp-Rh/Co/Re(0001) indicates that isolated skyrmions can be stabilized in the
ferromagnetic background with an applied magnetic field. Our results further
suggest that isolated skyrmions could be realized even in absence of an
external field in fcc-Rh/Co/Re(0001), hcp-Pd/Co/Re(0001), fcc-Pd/Co/Re(0001)
and fcc-Ru/Co/Re(0001). This makes ultrathin films composed of a 4$d$
transition-metal and a Co layer on Re(0001) promising candidates for the
search of isolated skyrmions.

††preprint: APS/123-QED

##  I Introduction

Skyrmion lattices were experimentally discovered at low temperature in bulk
systems [1, 2] and at surfaces [3]. Subsequently, isolated magnetic skyrmions
[4] were observed in ultrathin films [5] and in magnetic multilayers [6, 7].
Since their discovery, skyrmions have been envisioned for many applications
[8, 9] ranging from next-generation logic devices [10, 11] and race-track
memories [11] to quantum computing [12, 13] as well as unconventional
computing such as probabilistic [14], neuromorphic [15] and reservoir
computers [16]. From an application point of view, transition-metal based
ultrathin films and multilayers are particularly promising in search for novel
skyrmion hosting systems with target properties since the magnetic
interactions, which stabilize skyrmions, can be tuned via interface structure
and composition [17, 18, 19, 20].

Transition-metal films on heavy metal surfaces serve as model systems to
uncover the underlying microscopic mechanisms related to skyrmions. The
interfacial Dzyaloshinskii-Moriya interaction (DMI) [21, 22], arises from
inversion symmetry breaking at the interface and large spin-orbit coupling due
to the heavy metal. It was first observed in a Mn monolayer on the W(110)
surface [23]. The discovery of a nanoskyrmion lattice in an Fe monlayer on
Ir(111) [3] marks the beginning of exploring skyrmions at the interface.
Stabilization of isolated skyrmions at the interface results from the
interplay of DMI along with the Heisenberg pairwise exchange interaction and
magnetocrystalline anisotropy.

In contrast to investigations of skyrmions in Fe-based transition-metal
ultrathin films [3, 5, 24, 25, 26, 27, 28, 29, 30], considerably less work
have been done to date on Co-based ultrathin films. Nonetheless, is it evident
from the existing reports that the Co-based films can host complex magnetic
structures including skyrmions [31, 32, 33, 34]. In pseudomorphycally grown
thin films of a Co monolayer and an atomic bilayer of Pt/Co on the Ir(111)
surface, Néel-type domain walls with a clockwise rotational sense were
observed in spin-polarized scanning tunneling microscopy (SP-STM) measurements
at low temperature [32]. The first reported Co-based ultrathin film which
hosts isolated skyrmions is Co/Ru(0001) [33]. The individual skyrmions in this
film system are (meta)stabilized at a low magnetic field of 150 mT with a
diameter of nearly 50 nm due to a competition between the Heisenberg exchange
interaction and DMI. Isolated zero-field skyrmions with diameters of about 5
nm have been observed in a Rh/Co atomic bilayer on the Ir(111) surface [34].
Large lifetimes have been predicted for skyrmions in Rh/Co/Ir(111) due to the
frustration of exchange interactions [35]. Even antiskyrmions have been
proposed to be extremely stable in this film system with calculated lifetimes
[35] similar to those obtained for skyrmions in Pd/Fe/Ir(111) [28, 36, 37].

Recently, a variety of complex magnetic structures has been found in
transition-metal monolayers and transition/heavy metal bilayers deposited on
the Re(0001) surface [38, 39, 40, 41]. These observations underscore the
potential of this substrate as a promising candidate for realizing novel
magnetic structures including skyrmions, alongside the well-known Ir and Rh
substrates. The first experimental realization of the row-wise
antiferromagnetic state and the 3$Q$ state, predicted theoretically nearly 20
years ago [42], occurs in a Mn monolayer on Re(0001) [38]. Subsequently, the
3Q state has also been observed in Pd/Mn and Rh/Mn bilayers on Re(0001)
through SP-STM studies [41]. Theoretical studies have demonstrated that the
isolated skyrmions can be stabilized in a Pd/Fe bilayer on Re(0001) with the
aid of an external magnetic field [39, 43]. Moreover, theoretical
investigations have highlighted the importance of the complex noncollinear
structures in triggering topological superconductivity when placed in
proximity to a conventional superconductor [44]. Experimentally, nanoscale
skyrmions have been realized in Pd/Fe bilayers grown on a few atomic layers of
Ir on the Re(0001) surface [45].

Here, we study the magnetic interactions in atomic bilayers, consist of a Co
and a $4d$ transition metal (Ru, Rh, Pd) layers, on the Re(0001) surface using
the ab-initio density functional theory (DFT) methods. We consider different
compositions, i.e., a Co/$4d$ and a $4d$/Co bilayers, and fcc and hcp stacking
of the overlayer (Fig. 1). As a reference system, we include a Co monolayer in
hcp stacking on Re(0001). We perform DFT calculations for various collinear
and noncollinear magnetic states including the effect of spin-orbit coupling
(SOC). The goal of our investigation is to identify systems which are
promising for the realization of topological spin structures such as skyrmions
and antiskyrmions.

Based on our DFT calculations, we show that the bilayer hybridization changes
significantly with the variation of bilayer composition and symmetry of the
overlayer resulting in a change of spin spiral energy dispersion across the
films considered. We find a spin spiral ground state for hcp-Rh/Co/Re(0001)
with a period of 13 nm, while all other films exhibit a ferromagnetic ground
state. Interestingly, the energy dispersion indicates that isolated skyrmions
can be found in hcp-Rh/Co/Re(0001) by applying an external magnetic field. The
flat energy dispersion of spin spirals in the vicinity of the ferromagnetic
ground suggests that zero-field magnetic skyrmions can be realized in four of
the studied film systems: fcc-Rh/Co/Re(0001), hcp-Pd/Co/Re(0001), fcc-
Pd/Co/Re(0001) and fcc-Ru/Co/Re(0001).

Figure 1: Sketches of the ultrathin films considered in this study. Top row
shows four film systems with a Co overlayer: (a) Co (b) Co/Rh, (c) Co/Pd, and
(d) Co/Ru on Re(0001). Bottom row shows three films where the bilayer
composition is swapped: (e) Rh/Co (f) Pd/Co and (g) Ru/Co on Re(0001). For
each film, except for Co/Re(0001), hcp and fcc stacking orders of the
overlayer are considered which adds up to thirteen films.

We evaluate the exchange interaction and the DMI within the nearest-neighbor
approximation by mapping the total DFT energies to an effective atomistic spin
model. In addition, we calculate the magnetocrystalline anisotropy energy
(MAE) to identify the easy magnetization direction of the films. The spin
spiral dispersion of films with a Co overlayer is very steep and the effective
nearest-neighbor exchange constant is quite large. For systems in which the Co
layer is sandwiched between the $4d$ overlayer and the Re(0001) surface, the
dispersion is flatter close to the ferromagnetic state and the exchange
constant is much reduced. For selected film systems, we obtain exchange
constants beyond nearest-neighbors which allows future spin dynamics
simulations to verify the predicted occurrence of skyrmions. Given the
experimental evidence of a submonolayer pseudomorphic growth of Co on Re(0001)
[46], our proposed films present an intriguing prospect for further
experimental investigations.

This paper is organized in three sections. In the first section (section II),
we outline the methodology employed to compute the electronic structures and
magnetic properties of ultrathin films and provide details of the
calculations. In the second section (section III), we present and analyze the
results of all thirteen ultrathin films, as well as provide an in-depth
discussion. In the final section, section IV, we offer a comprehensive summary
of the investigation.

##  II  Computational details

To compute the electronic and magnetic structures of ultrathin films, we used
two of the most accurate first-principles DFT methods, namely, the projected
augmented wave (PAW) method as implemented in the vasp code [47] and the full-
potential linearized augmented plane-wave (FLAPW) method as implemented in the
fleur code [48]. Various preceding reports have shown the accuracy and
efficiency of these two DFT codes in computing the electronic structures and
magnetic properties of ultrathin transition-metal films [3, 49, 50, 51, 39].

We used the vasp code to perform structural relaxations of the films in
collinear magnetic states. In line with previous works [39, 52, 43, 40], we
have chosen the in-plane lattice constant ($a_{in}$) of the film as the
experimental lattice constant of bulk Re, $a$= $2.761$ Å, while the interlayer
distance ($d_{0}$) of the films was set to the bulk layer separation $c/a$=
$2.228$ Å [53]. We modeled the experimental film geometry as an asymmetric
film comprising of a monolayer or a bilayer on top of a nine layers Re(0001)
substrate. To obtain the equilibrium geometry of these films, we allowed the
top three layers (two for hcp-Co/Re(0001)) to relax along the $z$-direction
via minimizing the forces on all atoms to less than $0.01$ eV/Å using the vasp
code. We used the generalized gradient approximation (GGA) as parameterized by
Perdew, Bruke and Ernzerhof [54] for the exchange-correlation part of the
potential. The energy cut-off was set to 500 eV and we used 66 $k$ points in
the irreducible part of the two-dimensional Brillouin zone (2DBZ) to perform
the BZ integration.

The magnetic properties of the structurally optimized films were obtained
using the fleur code. In particular, we calculated the energy dispersion
$E(\mathbf{q})$ of homogeneous flat spin spirals self-consistently in the
scalar-relativistic approximation [55] as a function of wave vector
$\mathbf{q}$. The magnetic moment $\textbf{M}_{i}$ at the atomic site
$\textbf{R}_{i}$ of a spin spiral is specified by
$\textbf{M}_{i}$=$M\left(\textrm{cos}(\textrm{{q}}\cdot\textrm{{R}}_{i}),\textrm{sin}(\textrm{{q}}\cdot\textrm{{R}}_{i}),0\right)$,
where $M$ is the magnitude of the moment. For spin spiral calculations, we
used the local density approximation (LDA) of the exchange-correlation
potential, as parameterized by Vosko, Wilk and Nusair [56]. The total DFT
energy can be mapped to the Heisenberg Hamiltonian

$\displaystyle\mathcal{H}_{\rm ex}=-\sum_{ij}J_{ij}\
(\textbf{m}_{i}\cdot\textbf{m}_{j})$ (1)

to evaluate the exchange interaction constants $J_{ij}$ between magnetic
moments $\textbf{m}_{i}$ and $\textbf{m}_{j}$ at sites $i$ and $j$,
respectively. We used the generalized Bloch theorem to compute the energy
dispersion [55] which allows us to use the chemical unit cell avoiding the
large supercell calculations. A dense mesh of 48$\times$48 $k$-points in full
2DBZ was used to compute the full energy dispersion as well as around the
$\overline{\Gamma}$ point, $\mid\textbf{q}\mid\leq 0.1\times\frac{2\pi}{a}$.
To get a good convergence in hcp-Co/Re(0001), we used a $k$-point mesh of
120$\times$120\. We set the cut-off for the basis functions, $k_{\rm max}$, to
4.0 a.u.-1.

The DMI arises due to the concerted effect of inversion symmetry breaking at
the interface and SOC, which favors a certain sense of rotation (clockwise or
anticlockwise). Within the Fert-Levy model [57], the DMI Hamiltonian can be
expressed as

$\displaystyle\mathcal{H}_{\rm
DMI}=-\sum_{ij}\textbf{D}_{\textrm{eff}}\cdot(\textbf{m}_{i}\times\textbf{m}_{j})$
(2)

where $\textbf{D}_{\textrm{eff}}$ is the effective DMI vector. Since the DMI
energy is quite small compared to the total energy of the system, we evaluate
it within the first-order perturbation theory based on the self-consistent
spin spiral calculations [58, 59]. Due to the film’s inherent symmetry, the
DMI vectors lie in the plane of the interface and they are perpendicular to
the line joining $\textbf{m}_{i}$ and $\textbf{m}_{j}$, thus, favoring
cycloidal spin spirals. The positive (negative) sign of the DMI constant
indicates that a right-rotating (left-rotating), i.e., clockwise
(anticlockwise), spin spiral is preferred.

The second energy contribution arising from SOC is the MAE which was
calculated self-consistently using the second variation approach [60]. We
performed fully self-consistent SOC calculations for the ferromagnetic state
to obtain the energy difference between the out-of-plane ($\perp$) and in-
plane ($\parallel$) magnetization directions, since MAE is defined as
$K_{\mathrm{MAE}}$= $E_{\parallel}$ $-$ $E_{\perp}$. Therefore, a positive
(negative) value of $K_{\mathrm{MAE}}$ signifies an out-of-plane (an in-plane)
easy magnetization axis. To accurately calculate the in-plane and out-of-plane
total energy difference, we used 13 to 17 substrate layers. To evaluate the
MAE, we used the same $k$-point mesh and value of $k_{\rm max}$ as for the
spin spiral calculations.

Table 1: Structural relaxed parameters for a Co monolayer and for atomic Co/4$d$ and 4$d$/Co bilayers on Re(0001). The interlayer distances and magnetic moments of the three outermost layers at their relaxed geometry in the FM state are given. The calculations are done with the vasp code using the GGA functional. The energy difference between the AFM and FM state ($E_{\mathrm{{AFM}}}-E_{\mathrm{{FM}}}$) is also shown. Systems |  |  $\Delta d_{1}$ (%)
---
|  $\Delta d_{2}$ (%)
---
|  $\Delta d_{3}$ (%)
---
| $\mu_{1}$
---
$(\mu_{\mathrm{B}})$
| $\mu_{2}$
---
$(\mu_{\mathrm{B}})$
| $\mu_{3}$
---
$(\mu_{\mathrm{B}})$
| $E_{\mathrm{{AFM}}}-E_{\mathrm{{FM}}}$
---
(meV/Fe)
hcp-Co/Re(0001) |  |  $-$12.93
---
| 0.97
---
| —
---
| 1.28
---
|  $-$0.06
---
| —
---
| 110.74
---
hcp-Co/Rh/Re(0001) |  |  $-$9.34
---
|  $-$3.05
---
| 0.09
---
| 1.93
---
| 0.28
---
|  $-$0.08
---
| 112.09
---
fcc-Co/Rh/Re(0001) |  |  $-$7.99
---
|  $-$1.71
---
|  $-$0.81
---
| 1.97
---
| 0.25
---
|  $-$0.08
---
| 111.50
---
hcp-Rh/Co/Re(0001) |  |  $-$11.13
---
|  $-$8.89
---
| 0.54
---
| 0.67
---
| 1.57
---
|  $-$0.08
---
| 111.57
---
fcc-Rh/Co/Re(0001) |  |  $-$10.23
---
|  $-$6.64
---
|  $-$0.36
---
| 0.73
---
| 1.67
---
|  $-$0.09
---
| 111.47
---
hcp-Co/Pd/Re(0001) |  |  $-$6.30
---
| 2.03
---
|  $-$2.12
---
| 1.99
---
| 0.16
---
|  $-$0.02
---
| 212.78
---
fcc-Co/Pd/Re(0001) |  |  $-$5.93
---
| 2.50
---
|  $-$2.37
---
| 1.99
---
| 0.14
---
|  $-$0.02
---
| 208.02
---
hcp-Pd/Co/Re(0001) |  |  $-$7.09
---
|  $-$10.23
---
| 0.54
---
| 0.25
---
| 1.51
---
|  $-$0.08
---
| 69.04
---
fcc-Pd/Co/Re(0001) |  |  $-$5.79
---
|  $-$10.31
---
| 0.96
---
| 0.21
---
| 1.48
---
|  $-$0.08
---
| 55.22
---
hcp-Co/Ru/Re(0001) |  |  $-$12.48
---
|  $-$4.40
---
| 1.89
---
| 1.76
---
| 0.16
---
|  $-$0.08
---
| 90.26
---
fcc-Co/Ru/Re(0001) |  |  $-$9.34
---
|  $-$3.50
---
| 0.98
---
| 1.88
---
| 0.29
---
|  $-$0.06
---
| 138.91
---
hcp-Ru/Co/Re(0001) |  |  $-$12.93
---
|  $-$8.44
---
| 0.54
---
| 0.34
---
| 1.21
---
|  $-$0.06
---
| 55.72
---
fcc-Ru/Co/Re(0001) |  |  $-$11.58
---
|  $-$6.19
---
|  $-$0.35
---
| 0.61
---
| 1.49
---
|  $-$0.03
---
| 60.41
---

##  III  Results and discussion

###  III.1  Structural parameters and collinear magnetism

We start by presenting our results of two collinear configurations, i.e., the
ferromagnetic (FM) and the row-wise antiferromagnetic (AFM) state, for all
thirteen films in Table 1, as computed by the vasp code using the GGA
exchange-correlation potential.

To obtain the collinear magnetic state of lowest energy, we first relax the
topmost three layers for all twelve films except hcp-Co/Re(0001), where only
the two topmost layers, i.e., the Co overlayer and the first substrate Re
layer are relaxed, perpendicular to the plane in the FM and AFM
configurations, keeping all the other substrate layers fixed. We calculate the
energy difference $\Delta E$=$E_{\mathrm{AFM}}-E_{\mathrm{FM}}$ for all films
from the total energy of the FM and AFM configurations evaluated at their
respective relaxed geometry.

The energy difference $\Delta E$ is positive for all films (last column of
Table 1) which illustrates that all systems prefer FM alignment of spins.
However, the value of $\Delta E$ varies significantly among the films from 55
meV/Co atom for fcc-Pd/Co/Re(0001) to 212 meV/Co atom for hcp-Co/Pd/Re(0001),
which indicates that the latter film is a stronger ferromagnet than the former
one. We observe that the energy difference $\Delta E$ is quite small for Pd/Co
and Ru/Co bilayers, i.e., when the Co layer is sandwiched between the 4$d$
transition metal and Re substrate, as compared to Co/Pd and Co/Ru bilayers,
i.e., when Co is an overlayer. Interestingly, the energy difference $\Delta E$
is similar for Co/Rh and Rh/Co bilayers.

The optimized interlayer distances and the magnetic moments of the topmost
three layers (two for hcp-Co/Re(0001)) are listed for the FM state in Table 1.
We define the relaxation of the interlayer distances relative to the bulk
value as follows

$\displaystyle\Delta
d_{\mathrm{i}}(\%)=\left(\frac{d_{\mathrm{i,i+1}}-d_{0}}{d_{0}}\right)\times
100$ (3)

where $d_{\mathrm{i,i+1}}$ is the relaxed interlayer distance between layer
$i$ and $i+1$, and $d_{0}$ is the bulk interlayer distance of Re, i.e., 2.228
Å.

The relaxation of the overlayer is inward for all films and the first relaxed
interlayer distance ($\Delta d_{1}$) is fairly large as compared to the second
($\Delta d_{2}$) and third ($\Delta d_{3}$) ones. However, the hcp and fcc
stacked Pd films, i.e., hcp-Pd/Co/Re(0001) and fcc-Pd/Co/Re(0001), behave in a
different manner. The relaxed second interlayer distance ($\Delta d_{2}$) of
the two Pd/Co/Re(0001) films is larger than the first one ($\Delta d_{1}$).
$\Delta d_{1}$ varies from $-5.8\%$ for fcc-Pd/Co/Re(0001) to $-12.9\%$ for
hcp-Co/Re(0001) and hcp-Ru/Co/Re(0001). We observe a variation of the first
interlayer distance ($\Delta d_{1}$) by more than $3\%$ when the stacking
order of the overlayer Co is changed from hcp to fcc in Co/Ru/Re(0001) films.

The second interlayer relaxation is also inward for most films, except for
both Co/Pd/Re(0001) films, which exhibit an outward relaxation. The value of
the relaxed second interlayer distance ($\Delta d_{2}$) is significantly
smaller for all films as compared to the first interlayer distance. An
exception is the Pd/Co/Re(0001) films, where the second interlayer distance is
considerably larger than the first one.

The third interlayer spacing ($\Delta d_{3}$) is, in general, one order of
magnitude smaller than the second one. However, for Co/Pd and Co/Ru bilayers
on Re(0001), it is comparable to the second interlayer distance. The three
relaxed interlayer spacing indicate that they are strongly dependent on the
composition of bilayers and the stacking symmetry of the overlayer. For
example, a change in the stacking symmetry of the toplayer from hcp to fcc
turns the third interlayer relaxation for Rh/Co and Ru/Co bilayers on Re(0001)
from outward to inward.

We notice a similar trend of relaxed interlayer distances in previously
reported fcc and hcp stacked Rh overlayers on Co/Ir(111) [34]. The relaxation
of the topmost Rh and sandwiched Co layers are inward and the first relaxed
interlayer distance ($\Delta d_{\textrm{Rh}-\textrm{Co}}$) is significantly
larger than the second one ($\Delta d_{\textrm{Co}-\textrm{Ir}}$). The third
interlayer relaxation ($\Delta d_{\textrm{Ir}-\textrm{Ir}}$) is outward and
its magnitude is larger than the second interlayer distance.

The magnetic moment of the Co overlayer varies only little for Co/Rh, Co/Pd
and Co/Ru bilayers on Re(0001), i.e., between 1.8 $\mu_{\mathrm{B}}$ and 1.9
$\mu_{\mathrm{B}}$. It is reduced to 1.3 $\mu_{\mathrm{B}}$ for hcp-
Co/Re(0001). The magnetic moment is also reduced for the Co layer, when it is
sandwiched between the 4$d$ transition metal and Re substrate. It is about 1.6
to 1.7 $\mu_{\mathrm{B}}$ for Rh/Co bilayers, however, for fcc-Ru/Co on
Re(0001), the value is much lower, i.e., 1.2 $\mu_{\mathrm{B}}$. The Co layer
also induces a magnetic moment to the adjacent 4$d$ transition-metal layer and
the first Re substrate layers which are aligned parallel and antiparallel to
the Co moment, respectively. The value of the magnetic moment of 4$d$
transition-metal layer is one order of magnitude smaller than of the Co
moment, whereas it is two order of magnitude smaller, i.e., nearly negligible,
for the first Re substrate layer.

###  III.2  Noncollinear magnetism and effective spin model

In the preceding section, we examined the two collinear spin structures,
namely, the FM and AFM states, and found that the lowest energy state of all
the films is the FM spin configuration. In this section, we explore the
existence of noncollinear ground state in these film systems taking the effect
of SOC into account, i.e., energy contributions due to DMI and MAE. Below we
present the energy dispersion $E(\mathbf{q})$ of homogeneous cycloidal spin
spirals for all considered ultrathin films (Fig. 1) as a function of wave
vector q including DMI and MAE calculated using the fleur code in the vicinity
of the $\overline{\Gamma}$ point ($\mid$q$\mid\leq 0.1\times\frac{2\pi}{a}$),
which represents the FM state, along two high-symmetry directions
$\overline{\Gamma\mathrm{KM}}$ and $\overline{\Gamma\mathrm{M}}$ of the 2DBZ.

Figure 2: (a) Energy dispersion $E$(q) of homogeneous left-rotating
(anticlockwise) cycloidal spin spirals for hcp-Co/Re(0001) in the vicinity of
the $\overline{\Gamma}$ point (FM state) along the
$\overline{\Gamma\mathrm{KM}}$ and $\overline{\Gamma\mathrm{M}}$ directions of
2DBZ. Filled circles present DFT data including SOC, i.e., DMI and MAE, and
the solid lines are fit to the effective Heisenberg and DMI spin models. Note
that the MAE shifts all spin spiral energies along the positive energy axis by
$K_{\textrm{MAE}}$/2 with respect to the collinear FM state at zero energy.
(b) Total energy contribution due to DMI (red) and individual contribution of
the Co layer (purple) and the Re substrate (brown). Filled circles present DFT
data and solid lines are fit to the effective DMI model (Eq. 2). The DFT
calculations are done with the fleur code.

Conditions for observing skyrmions in an ultrathin film are either (i) it
shows a shallow spin spiral energy minimum close to the FM state
($\overline{\Gamma}$ point) or (ii) it has a FM ground state, but exhibits a
flat energy dispersion $E(\mathbf{q})$ close to the $\overline{\Gamma}$ point.
In the first case, isolated skyrmions can be stabilized in the field-polarized
phase by applying an external magnetic field at a minimum value that
compensates the energy depth of the spin spiral minimum. An example for this
case is the Pd/Fe bilayer on Ir(111) [5, 25, 61]. The second case is
particularly intriguing since zero-field skyrmions can be stabilized under
this condition in the ferromagnetic ground state. Recently, isolated skyrmions
in the absence of an external field have been realized in Rh/Co bilayers on
Ir(111) [62] meeting this condition and also a coexistence of nanoscale
skyrmions and antiskyrmions has been predicted in this film [35].

For these reasons, we only display the energy dispersions up to a value of $q$
$\approx 0.1\times\frac{2\pi}{a}$ on both sides of the $\overline{\Gamma}$
point along $\overline{\Gamma\mathrm{KM}}$ and $\overline{\Gamma\mathrm{M}}$
directions in this section. We restrict the dispersion curve with a primary
objective to identify the promising candidates among the thirteen films which
can host skyrmions. In addition, our systematic study allows us to gain
insight into the effect of bilayer composition and their associated symmetries
on the effective magnetic interactions and hence on the magnetic ground state.
In section III.4, we display the full dispersion curve for selected promising
films and give a parameterization of an atomistic spin model containing beyond
nearest-neighbour exchange constants required to accurately compute skyrmion-
related properties using atomistic spin dynamics simulations [61].

Fig. 2(a) shows such an energy dispersion for the reference system of a Co
monolayer in hcp stacking on the Re(0001) surface. The dispersion displays a
parabolic shape, i.e., a $q^{2}$ dependence, close to the $\overline{\Gamma}$
point, characteristic of a ferromagnetic film. By fitting the DFT total
energies obtained without SOC to Eq. 1, an effective nearest-neighbor exchange
constant $J_{\rm eff}$ can be obtained (Table 2). In the limit of a long
period, i.e., small value of $q$, a spin spiral state is energetically
unfavorable than the FM state by $K_{\rm MAE}/2$ per spin due to the
magnetocrystalline anisotropy energy (MAE). Therefore, the energy dispersion
in Fig. 2(a) is shifted by $K_{\rm MAE}/2$.

Due to symmetry, the DMI favors cycloidal spin spirals in ultrathin films. The
total DMI energy contribution to the dispersion of cycloidal spin spirals
(Fig. 2(b)) shows the characteristic linear dependence with $q$ in the
vicinity of the $\overline{\Gamma}$ point. A fit in this linear regime allows
to obtain the effective (nearest-neighbor) DMI constant, $D_{\rm eff}$. The
energy contribution due to DMI is quite small at $q$
$=0.1\times\frac{2\pi}{a}$, i.e., only about 0.2 meV/Co atom, leading to a
$D_{\rm eff}$ of $-$0.12 meV. The negative sign of $D_{\rm eff}$ (positive
energy contribution in Fig. 2(b)) indicates a favorable anticlockwise
rotational sense. In our DFT calculation, we can decompose the total energy
contributions from DMI with respect to the different layers [58]. As seen in
Fig. 2(b), the DMI energies due to Re and Co layers exhibit opposite signs,
i.e., opposite rotational sense, and the Co contribution dominates.

Table 2: Effective interaction constants for ultrathin films composed of Co/4$d$ and 4$d$/Co bilayers on Re(0001) (Fig. 1). The effective exchange ($J_{\mathrm{eff}}$) and the Dzyaloshinskii–Moriya ($D_{\mathrm{eff}}$) interactions constants are obtained from the fitting to total energy close to the $\overline{\Gamma}$ point ($\mid$q$\mid\leq 0.1\times\frac{2\pi}{a}$). $K_{\mathrm{MAE}}$ is calculated directly with the fleur code. A positive (negative) DMI constant indicates that the right-rotating (left-rotating), i.e., clockwise (anticlockwise) spin spirals are favored. A negative (positive) $K_{\mathrm{MAE}}$ indicates an in-plane (out-of-plane) easy axis. The magnetic order of the ground state is also indicated. FM∗ indicates potential film for zero-field skyrmions. Systems |  | $J_{\mathrm{eff}}$
---
(meV)
| $D_{\mathrm{eff}}$
---
(meV)
| $K_{\mathrm{MAE}}$
---
(meV)
| Ground
---
state
hcp-Co/Re(0001) |  | 17.50
---
|  $-$0.12
---
| 0.40
---
| FM
---
hcp-Co/Rh/Re(0001) |  | 25.70
---
| 0.12
---
|  $-$1.16
---
| FM
---
fcc-Co/Rh/Re(0001) |  | 18.02
---
| 0.16
---
| 0.54
---
| FM
---
hcp-Rh/Co/Re(0001) |  | 14.10
---
| 1.91
---
| 0.10
---
| SS
---
fcc-Rh/Co/Re(0001) |  | 12.67
---
| 1.17
---
| 0.60
---
| FM∗
---
hcp-Co/Pd/Re(0001) |  | 27.40
---
| 0.27
---
| 1.98
---
| FM
---
fcc-Co/Pd/Re(0001) |  | 26.51
---
| 0.14
---
| 1.98
---
| FM
---
hcp-Pd/Co/Re(0001) |  | 15.12
---
| 0.70
---
| 0.14
---
| FM∗
---
fcc-Pd/Co/Re(0001) |  | 5.47
---
| 0.58
---
| 0.20
---
| FM∗
---
hcp-Co/Ru/Re(0001) |  | 13.73
---
|  $-$0.32
---
|  $-$1.71
---
| FM
---
fcc-Co/Ru/Re(0001) |  | 18.05
---
| 0.07
---
| 1.23
---
| FM
---
hcp-Ru/Co/Re(0001) |  | 18.10
---
| 0.11
---
| 0.37
---
| FM
---
fcc-Ru/Co/Re(0001) |  | 6.90
---
| 0.07
---
| 0.11
---
| FM∗
---
Figure 3: Energy dispersion $E$(q) of homogeneous right-rotating (clockwise)
cycloidal spin spirals around the $\overline{\Gamma}$ point along
$\overline{\Gamma\mathrm{KM}}$ and $\overline{\Gamma\mathrm{M}}$ directions of
2DBZ for (a) hcp-Co and fcc-Co on Rh/Re(0001) and (d) hcp-Rh and fcc-Rh on
Co/Re(0001). Filled circles present DFT data including SOC, i.e., DMI and MAE,
and the solid lines are fit to the effective Heisenberg and DMI spin models.
Note that the MAE shifts all spin spiral energies along the positive energy
axis by $K_{\textrm{MAE}}$/2 with respect to the collinear FM state at zero
energy. (b,c,e,f) Total energy contribution due to DMI (red for hcp-Co/Rh and
hcp-Rh/Co and green for fcc-Co/Rh and fcc-Rh/Co) and individual contribution
of the Co (purple ) and Rh (yellow) layers and the Re substrate (brown).
Filled circles present DFT data and solid lines are fit to the effective DMI
model (Eq. 2). The DFT calculations are done with the fleur code. Figure 4:
Energy dispersion $E$(q) of homogeneous right-rotating (clockwise) cycloidal
spin spirals around the $\overline{\Gamma}$ point along
$\overline{\Gamma\mathrm{KM}}$ and $\overline{\Gamma\mathrm{M}}$ directions of
2DBZ for (a) hcp-Co and fcc-Co on Pd/Re(0001) and (d) hcp-Pd and fcc-Pd on
Co/Re(0001). Filled circles present DFT data including SOC, i.e., DMI and MAE,
and the solid lines are fit to the effective Heisenberg and DMI models. Note
that the MAE shifts all spin spiral energies along the positive energy axis by
$K_{\textrm{MAE}}$/2 with respect to the collinear FM state at zero energy.
(b,c,e,f) Total energy contribution due to DMI (red for hcp-Co/Pd and hcp-
Pd/Co and green for fcc-Co/Pd and fcc-Pd/Co) and individual contribution of
the Co (purple) and Pd (yellow) layers and the Re substrate (brown). Filled
circles present DFT data and solid lines are fit to the effective DMI model
(Eq. 2). The DFT calculations are done with the fleur code. Figure 5: Energy
dispersion $E$(q) of homogeneous left-rotating (anticlockwise) cycloidal spin
spirals for (a) hcp-Co/Ru/Re(0001) and right-rotating (clockwise) cycloidal
spin spirals for (a) fcc-Co/Ru/Re(0001) and (d) hcp-Ru and fcc-Ru on
Co/Re(0001) around the $\overline{\Gamma}$ point along
$\overline{\Gamma\mathrm{KM}}$ and $\overline{\Gamma\mathrm{M}}$ directions of
2DBZ. Filled circles present DFT data including SOC, i.e., DMI and MAE, and
the solid lines are fit to the effective Heisenberg and DMI models. Note that
the MAE shifts all spin spiral energies along the positive energy axis by
$K_{\textrm{MAE}}$/2 with respect to the collinear FM state at zero energy.
(b,c,e,f) Total energy contribution due to DMI (red for hcp-Co/Ru and hcp-
Ru/Co and green for fcc-Co/Ru and fcc-Ru/Co) and individual contribution of
the Co (purple) and Ru (yellow) layers and the Re substrate (brown). Filled
circles present DFT data and solid lines are fit to the effective DMI model
(Eq. 2). The DFT calculations are done with the fleur code.

Fig. 3(a,d) show the energy dispersion curves of spin spirals for Co/Rh and
Rh/Co bilayers on Re(0001) for both fcc and hcp stacking of the topmost layer.
Note that these dispersions include the effect of SOC, i.e., contribution of
the DMI and the MAE, and display the energy of cycloidal spin spirals with a
clockwise rotational sense favored by DMI. We find that the curves for the
films with a Co overlayer (Fig. 3(a)) are qualitatively similar to that for
the Co monolayer on Re(0001) (Fig. 2(a)). For both stackings of the Co
overlayer, we find a ferromagnetic ground state and a parabolic energy
dispersion. The value of the effective exchange constant is considerably
larger for the hcp-Co/Rh bilayer than for the fcc-Co/Rh bilayer or the Co
monolayer on Re(0001) (Table 2).

For the Rh/Co bilayers (Fig. 3(d)), in which the Co layer is sandwiched
between the Rh layer and the Re substrate, the energy dispersions look
qualitatively different. Both curves display local energy minima for finite
values of $q$ close to the $\overline{\Gamma}$ point. For hcp stacking of the
Rh overlayer, a spin spiral energy minimum is obtained, while the
ferromagnetic state (at zero energy) remains slightly favorable for fcc
stacking. The minimum of the spin spirals in hcp-Rh/Co/Re(0001) occurs at
$q\approx$ 0.020 $\times\frac{2\pi}{a}$ with a pitch of $\lambda=$13.55 nm and
it is nearly 0.35 meV/Co atom lower than the FM state. This shallow
noncollinear minimum close to the FM state is similar to that observed in the
prototypical skyrmion system Pd/Fe/Ir(111) [61] and it implies that an
external magnetic field of about 3 Tesla can lead to the field-polarized phase
in which isolated skyrmions can be stabilized.

The lowest energy state of fcc-Rh/Co/Re(0001) (Fig. 3(d)) is ferromagnetic.
However, in contrast to the Co/Rh bilayers (Fig. 3(a)), the dispersion curves
are very flat in the vicinity of the $\overline{\Gamma}$ point and rise only
gradually as $q$ increases. A similar type of flat dispersion curve has been
reported for Rh/Co/Ir(111) [62] and it leads to the formation of isolated
skyrmions in absence of an external magnetic field as observed using SP-STM
studies [62, 63]. Therefore, we anticipate that zero-field skyrmions could
also occur in fcc-Rh/Co bilayer on Re(0001).

One reason for the different types of spin spiral energy dispersions obtained
for Co/Rh versus Rh/Co bilayers on Re(0001) is the considerably reduced
effective exchange interaction for the latter films (Table 2). Another reason
is the increase of the DMI constant by about one order of magnitude for the
Rh/Co bilayers as compared Co/Rh bilayers on Re(0001) (Table 2). This effect
is also clearly visible in the SOC contributions to the spin spiral
dispersions (Figs. 3(b,c) versus (e,f)). Note that a linear fit to the total
energy contribution due to SOC is excellent in the considered region close to
the $\overline{\Gamma}$ point.

The contributions from the Re substrate to the DMI energy is expected to be
large since it is a heavy $5d$ transition metal with a large SOC constant.
Interestingly, the Rh contribution is of similar magnitude as that of Re for
fcc-Co/Rh/Re(0001) (Fig. 3(c)) and even exceeds the Re contribution for hcp-
Rh/Co/Re(0001) (Fig. 3(e)). A detailed understanding of the contribution to
the DMI from different atomic layers at the interface requires an in-depth
analysis of the electronic structures which we shown in Ref. [20] for a few
systems.

For atomic bilayers consisting of a Co and a Pd layers, the general trend of
the spin spiral energy dispersions (Fig. 4) is similar to that the above Rh
bilayers. In particular, we find that for the films in which the Co layer is
at the surface, i.e., Co/Pd/Re(0001) (Fig. 4(a)), $E(\mathbf{q})$ rises quite
steeply in the vicinity of the $\overline{\Gamma}$ point, characteristic of a
strongly ferromagnetic film. Accordingly, the values of $J_{\rm eff}$ (Table
2) are on the order of that for hcp-Co/Rh/Re(0001). In contrast, the Pd/Co
bilayers on Re(0001) (Fig. 4(d)) display a rather flat energy dispersion which
indicates significant frustration of exchange interactions. This is reflected
in the relatively small effective nearest-neighbor exchange constants (Table
2).

The DMI energy contribution to spin spirals is much enhanced for the
sandwiched structures (Figs. 4(e,f)) compared to films with a Co overlayer
(Figs. 4(b,c)). However, the effective DMI constant is smaller than that
obtained for Rh/Co/Re(0001) (Table 2). Therefore, we do not find spin spiral
energy minima for the Pd/Co bilayers on Re(0001), but observe a very shallow
rise of the energy dispersion $E(\mathbf{q})$ (Fig. 4(d)). Concerning the
possible formation of skyrmions, the ultrathin Pd/Co film with a fcc stacking
of the Pd layer, i.e., fcc-Pd/Co/Re(0001) (Fig. 4(d)), is most promising due
to the very slow rise of the energy of spin spirals with $\mathbf{q}$. On the
other hand, the energy dispersion rises more quickly close to the
$\overline{\Gamma}$ point for hcp-Pd/Co/Re(0001), but the DMI is also
increased (Table 2) as compared to fcc-Pd/Co/Re(0001). Since these films
exhibits a ferromagnetic ground state and a flat energy dispersion,
metastable, individual skyrmions might even appear without an applied magnetic
field similar to observations in Rh/Co/Ir(111) [62].

Finally, we turn to the bilayers consisting of a Ru and a Co layers on
Re(0001) (Fig. 5). In these films, the overall trend is similar to that
observed for the Pd or Rh bilayers (Figs. 3 and 4, respectively). In
particular, the energy dispersion rises quickly for Co/Ru/Re(0001) films (Fig.
5(a)) and the effective exchange constants are relatively large (Table 2). For
the sandwich structured hcp-Ru/Co/Re(0001) (Fig. 5(d)), the energy dispersion
rises almost as quickly as for the Co/Ru/Re(0001) film systems. But for fcc-
Ru/Co/Re(0001) (Fig. 5(d)), we observe a much flatter energy dispersion with a
small exchange constant $J_{\rm eff}$ of similar magnitude as for fcc-
Pd/Co/Re(0001). However, the DMI, which favors clockwise cycloidal spin
spirals in Ru/Co/Re(0001), is much smaller than the Co films with a Pd
overlayer (Table 2). Therefore, the energy dispersion of spin spirals is not
as shallow as for the fcc-Pd/Co bilayer. Thus we infer that the formation of
isolated skyrmions in the ferromagnetic background might be possible for fcc-
Ru/Co/Re(0001), but less favorable than the films with Pd/Co bilayers.

Table 2 summarizes and quantifies the main results from the analysis of the
spin spiral energy dispersions. The effective exchange $J_{\mathrm{eff}}$ and
DMI $D_{\mathrm{eff}}$ constants for all thirteen films are listed, which are
computed by fitting the spin spiral dispersions without and including SOC,
respectively, in the vicinity of the $\overline{\Gamma}$ point within the
nearest-neighbor approximation. In addition, the MAE and the spin structure of
the ground state are given.

The effective exchange constant ($J_{\mathrm{eff}}$) provides valuable
insights into the behavior of the dispersion curve within a narrow region
around the $\overline{\Gamma}$ point. Precisely, it describes the curvature of
the dispersion close to the $\overline{\Gamma}$ point. The positive value of
the exchange constants indicates that the ground state of all the films is
ferromagnetic when SOC was neglected. Moreover, the magnitude of the constant
reflects the strength of the ferromagnetic interaction. Therefore, we can
infer that hcp-Co/Pd/Re(0001) is the strongest ferromagnet and fcc-
Pd/Co/Re(0001) is the weakest. We observe that the exchange constant of the
films where Co is an overlayer is larger compared to the films with same
composition but Co becomes a sandwiched layer. However, an exception is
observed for hcp stacked Ru films. As the bilayer composition is flipped from
hcp-Co/Ru to hcp-Ru/Co, instead of reducing, the effective exchange constant
increases by 4.37 meV.

The DMI energy is positive for hcp-Co/Re(0001) and hcp-Co/Ru/Re(0001) and
negative for all other films. The positive (negative) value of DMI indicates
that right-rotating (left-rotating), i.e., clockwise (anticlockwise) cycloidal
spin spirals are favored. The magnitude of DMI constant is largest for hcp-
Rh/Co/Re(0001), i.e., 1.91 meV and smallest for fcc-Co/Ru/Re(0001) and fcc-
Co/Ru/Re(0001), i.e., 0.07 meV.

Table 2 also contains the MAE constants for all films under consideration. The
magnitude of the MAE is larger for films with a Co overlayer as compared to
films with a sandwiched Co layer due to the reduced coordination at the
surface. The MAE attains a maximum value for hcp and fcc-Co/Pd/Re(0001) which
is 1.98 meV/Co atom and the lowest value is 0.10 meV/Co atom obtained for hcp-
Rh/Co/Re(0001). The MAE of hcp-Co/Rh/Re(0001) and hcp-Co/Ru/Re(0001) is
negative and thus favors an in-plane easy magnetization direction. For all
other films, an out-of-plane easy axis is obtained which favors the formation
of skyrmions.

It is mentioned in the previous section that a large value of
$J_{\mathrm{eff}}$ indicates a faster rise of $E(\mathbf{q})$ (without SOC)
around the FM state ($\overline{\Gamma}$ point). As the value of
$J_{\mathrm{eff}}$ is reduced, the dispersion curve becomes flatter. In such a
situation, a significant DMI constant above 1 meV makes the spin spiral states
more favorable than the FM state in hcp-Rh/Co and fcc-Rh/Co bilayers on
Re(0001) (Fig.3(d)). However, MAE acts opposite to the spin spiral
stabilization, completely compensates the spin spiral energy of fcc-
Rh/Co/Re(0001), leading to a FM ground state and partially compensates the
spin spiral energy of hcp-Rh/Co/Re(0001), leading to a spin spiral ground
state. Therefore, the stability of spin spiral states depends on the intricate
interplay among these three interactions.

Figure 6: Spin-polarized local density of states (LDOS) of the (a) Co, (b) Ru
and (c) the first Re layers in hcp-Co/Ru/Re(0001) (red curves) and in fcc-
Co/Ru/Re(0001) (blue curves). (d-f) Spin-resolved LDOS of the Co, Ru, and the
first Re layers, respectively, in hcp-Co/Ru/Re(0001) (green curves) and in
hcp-Ru/Co/Re(0001) (orange curves). The Fermi level is set to zero.

Finally, a comparison with ultrathin Co-based films on the Ir(111) surface can
be interesting. The effective exchange constant of hcp-Rh/Co and fcc-Rh/Co
bilayers on Ir(111) is about 6.0 and 5.4 meV [34], respectively, i.e., nearly
half of the value found for the Rh/Co/Re(0001) films. The reduction in the
interaction constant leads to a flatter dispersion curve for Rh/Co/Ir(111)
films around the $\overline{\Gamma}$ point compared to the Rh/Co/Re(0001)
films. A sizable DMI constant of about 1.6 meV for hcp-Rh/Co/Ir(111) and about
0.8 meV for fcc-Rh/Co/Ir(111) favors right-rotating spin spirals over the FM
state. Similar to the Rh/Co/Re(0001) films, due to a partial and complete
compensation of the spin spirals energy by MAE, a FM and spin spiral ground
states is observed in fcc-Rh/Co/Ir(111) and hcp-Rh/Co/Ir(111), respectively.

###  III.3  Electronic structures

In Table 2, we show the variation of the effective exchange interaction
($J_{\mathrm{eff}}$), the effective DMI ($D_{\mathrm{eff}}$) and the MAE
($K_{\mathrm{MAE}}$) constants across the films considered. The former two
interaction constants vary as the hybridization of the Co layer changes with
the adjacent $4d$ and $5d$ transition-metal layers, whereas the MAE constant
($K_{\mathrm{MAE}}$) is sensitive to changes of the density of states (DOS) at
the Fermi level. In this section, we demonstrate the effect of changing the
overlayer stacking order (fcc versus hcp) and swapping the bilayer composition
on the local DOS (LDOS). To explore this, we choose three films: a hcp and fcc
stacked Co monolayer on Ru/Re(0001) and the sandwiched structure hcp-
Ru/Co/Re(0001), which is promising for observing skyrmions. We choose Ru films
since the largest variation of LDOS is observed among these films. The change
in LDOS of Co, Ru, and the first Re substrate layer for a variation of the
overlayer stacking from hcp-Co to fcc-Co on Ru/Re(0001) are displayed in Fig.
6(a-c), respectively, and the LDOS of the three layers in the same order for
swapping of the bilayer composition from hcp-Co/Ru to hcp-Ru/Co on Re(0001)
are displayed in Figs. 6(d-f), respectively. The LDOS are evaluated in the FM
state using the fleur code.

Before pointing out the intricate details, we summarize the general features
of LDOS. Since the films consist of 3$d$, 4$d$ and 5$d$ transition-metal
elements, the LDOS are predominantly influenced by the $d$ orbitals. It is
noteworthy that the $d$ orbitals become more delocalized as one moves from
3$d$ to 4$d$ and 5$d$ elements, i.e., from Co, to Ru and Re, respectively. As
a result, the value of the Co LDOS is significantly higher than that of the
other two elements and the intensity decreases as one moves from Ru to Re.
Furthermore, the change in the LDOS of Co and Ru due to swapping of bilayer
elements is quite substantial as compared to the change upon varying the
overlayer stacking symmetry. This can be attributed to the reduced
hybridization in case of a Co overlayer which is adjacent to only the Ru layer
(Fig. 6(a)). In contrast, for the sandwiched structure hcp-Ru/Co/Re(0001), the
Co layer is affected by both the Ru and the adjacent Re layers (Fig. 6(d)). A
direct consequence of the increased hybridization is a much reduced magnetic
moment of the sandwiched Co layer, i.e., 1.21 $\mu_{\rm B}$, as compared to a
value of 1.76 $\mu_{\rm B}$ in case of a Co overlayer (see Table 1). The LDOS
of Co show a large exchange splitting of majority and minority states and the
induced spin-polarization on the Ru layer is clearly visible. The spin
majority and minority LDOS of Re are nearly symmetric resulting in a small
induced magnetic moment for all three films (Table 1).

We observe that as the stacking order of the overlayer Co varies from hcp-Co
to fcc-Co on Ru/Re(0001) (Fig. 6(a)), the LDOS of Co change notably.
Conversely, the effect of the variation in overlayer stacking on the LDOS of
Rh and the first Re layer is relatively modest (Fig. 6(b,c)). The spin
majority peak of Co at $-2$ eV for hcp-Co/Ru/Re(0001) shifts to $-2.3$ eV for
fcc-Co/Ru/Re(0001). The intensity of the latter peak increases and becomes
sharper than the former one. The spin minority peak of Co for hcp-
Co/Ru/Re(0001) at 0.6 eV above the Fermi level moves closer to the Fermi level
as the stacking of Co changes to fcc, and the intensity increases by a little
amount.

There is almost no effect on the LDOS of Re (Fig. 6(f)) upon changing the
bilayer composition from Co/Ru to Ru/Co. On the other hand, significant
differences are observed in the LDOS of Co (Fig. 6(d)) and Ru (Fig. 6(e)). The
spin majority LDOS of Co for the overlayer and sandwiched layers are quite
small above 1 eV. Below this energy, the LDOS for the sandwiched Co layer
increase faster than that of the Co overlayer, become higher in magnitude and
show a peak at 0.7 eV below the Fermi level. Thereafter, the intensity of the
LDOS of the Co overlayer increases and becomes comparable to the sandwiched
layer from the energy range of $-0.50$ eV to $-4$ eV, except around $-2$ eV.
Around that energy, the LDOS of the Co overlayer show a sharp peak while the
LDOS from the sandwich layer exhibit a valley. The spin minority LDOS of Co
for both layer positions are similar above 0.5 eV and follow each other. Below
that energy, the differences become prominent. The LDOS of the Co overlayer
show a small peak at 0.25 eV and a large double peak around 0.75 eV below the
Fermi level, while the intensity of the LDOS of the Co sandwich layer remains
fairly small. Below $-1$ eV, the intensity of the LDOS from Co overlayer
reduces and becomes lower than the LDOS of the sandwiched Co layer.

The intensity of spin majority LDOS of Ru for hcp-Ru/Co/Re(0001) remains
nearly same over the energy range, but for hcp-Co/Ru, it shows a two-peak
structure with a significantly higher intensity as compared to hcp-
Ru/Co/Re(0001). The peaks are located at the Fermi level and 0.75 eV below the
Fermi level. Beside these variations, the intensity of LDOS for these two
films are comparable. In contrast, the spin majority LDOS are quite different.
The intensity of LDOS originating from Ru overlayer is higher above $-2$ eV
and below that energy, it is comparable to the sandwiched Ru layer. We observe
a sharp peak in the LDOS from Ru overlayer about 0.25 eV accompanied by a
small peak below 1.5 eV of the Fermi level.

An comparison of the LDOS of the Co/Ru versus Ru/Co bilayers on Re(0001) shows
that there are large variations in the electronic structures. In particular,
changing the position of the Co layer from being the topmost layer to the
sandwiched geometry results in a much stronger hybridization due to the
interaction with the Ru and the Re layers. This leads to a reduced magnetic
moment and a large modification of the interaction constants, especially a
smaller effective exchange constant reflecting exchange frustration (Table 2).
In contrast, changing the stacking order of the overlayer from hcp to fcc
affects the LDOS less in accordance with the qualitatively similar spin spiral
energy dispersion (Fig. 5).

###  III.4  Full energy dispersion and beyond nearest-neighbor exchange
interactions

Table 3: Beyond nearest-neighbor exchange interactions. ith nearest-neighbor exchange interaction constants, $J_{i}$, are presented for three selected films, hcp-Rh/Co/Re(0001), fcc-Rh/Co/Re(0001), and fcc-Pd/Co/Re(0001) which are promising to host skyrmions and for hcp-Co/Re(0001) as a reference system. The exchange constants are given in meV. Systems | $J_{1}$ | $J_{2}$ | $J_{3}$ | $J_{4}$ | $J_{5}$ | $J_{6}$ | $J_{7}$ | $J_{8}$ | $J_{9}$ | $J_{10}$
---|---|---|---|---|---|---|---|---|---|---
hcp-Co/Re(0001) | 9.18 | 12.43 |  $-$0.88 |  $-$1.65 |  $-$0.03 | 0.92 |  $-$0.92 | 1.04 | 0.19 |  $-$0.24
hcp-Rh/Co/Re(0001) | 16.64 | 0.80 |  $-$0.21 |  $-$0.19 |  $-$0.01 |  $-$0.27 | 0.12 |  $-$0.03 |  $-$0.05 | 0.08
fcc-Rh/Co/Re(0001) | 15.72 | 0.43 | 0.25 | 0.03 |  $-$0.42 | — | — | — | — | —
fcc-Pd/Co/Re(0001) | 7.04 | 0.40 |  $-$0.32 | 0.12 |  $-$0.46 | — | — | — | — | —
Figure 7: Energy dispersion $E$(q) of homogeneous flat spin spirals without
SOC along two high-symmetry directions $\overline{\Gamma\mathrm{KM}}$ and
$\overline{\Gamma\mathrm{M}}$ of the 2DBZ for (a) hcp-Co/Re(0001) (red) and
hcp-Rh/Co/Re(0001) (blue), (b) fcc-Rh/Co/Re(0001) (green) and fcc-
Pd/Co/Re(0001) (yellow). The solid circles present DFT data and the solid
lines are fit to the beyond nearest-neighbor Heisenberg model (Eq. 1). The
dashed lines are the fit within the nearest-neighbor Heisenberg model
(effective spin model) discussed in section III.2

In section III.2, we displayed the energy dispersion of all Co/4$d$ and
4$d$/Co bilayers on Re(0001) from the $\overline{\Gamma}$ point to
$\mid$q$\mid\leq 0.1\times\frac{2\pi}{a}$ and identified five 4$d$/Co bilayer
films based on the characteristics of the dispersion curve as promising for
skyrmions. We further computed the magnetic interactions and parameterized an
effective spin model to get an insight into the variation of energy dispersion
curves.

The effective spin model works well in the vicinity of the FM state, allowing
us to select promising films without calculating the full dispersion curve,
which is computationally expensive. However, as the $q$ value increases, the
effective model starts to deviate from the DFT result as shown in Fig. 7, and
prompts the necessity to incorporate the beyond nearest-neighbor interactions
into the analysis. The beyond nearest-neighbor exchange terms do not
contradict the qualitative conclusion drawn from the effective spin model on
skyrmion formation. However, the beyond nearest-neighbor exchange interactions
modifies skyrmion properties quantitatively such as critical magnetic fields,
phase diagrams, radius, energy barriers and lifetime [61, 36].

In Fig. 7, we display the full energy dispersion of homogeneous spin spirals
as a function of wave vector q excluding the effects of SOC along two high-
symmetry directions $\overline{\Gamma\mathrm{KM}}$ and
$\overline{\Gamma\mathrm{M}}$ of the 2DBZ for selected films: hcp-Rh/Co (Fig.
7(a)), fcc-Rh/Co and fcc-Pd/Co (Fig. 7(b)) on Re(0001), which have potential
for skyrmion formation and hcp-Co/Re(0001) (Fig. 7(a)), which is the reference
system and not promising for skyrmions. The high symmetry points of the 2DBZ
represent special spin structures: the FM state at the $\overline{\Gamma}$
point, row-wise AFM state at the $\overline{\mathrm{M}}$ point, and the Néel
state at the $\overline{\mathrm{K}}$ point.

The energy dispersion curves $E(\mathbf{q})$ of hcp-Rh, fcc-Rh and fcc-Pd on
Co/Re(0001) are qualitatively similar. The spin spiral energy increases as one
moves from the $\overline{\Gamma}$ point along $\overline{\Gamma\mathrm{KM}}$
and attains a maximum value at the $\overline{\mathrm{K}}$ point and then
decrease gradually with a further increase of $q$. On the other hand, the
curves rise from the $\overline{\Gamma}$ point along
$\overline{\Gamma\mathrm{M}}$ reache a maximum value at the
$\overline{\mathrm{M}}$ point. The steepness of the dispersion curve of hcp-Rh
and fcc-Rh on Co/Re(0001) is comparable, while it is quite low for fcc-
Pd/Co/Re(0001). The energy difference between the AFM ($\overline{\mathrm{M}}$
point) and FM ($\overline{\Gamma}$ point) state is nearly 120 meV for hcp-Rh
and fcc-Rh on Co/Re(0001), whereas the value is reduced by a factor of 2 to
$\approx$ 60 meV in fcc-Pd/Co/Re(0001). Note that the dispersion curve of
Rh/Co/Ir(111) which hosts zero-field skyrmions [34] behaves similar to that of
Rh/Co/Re(0001). The energy of the AFM state with respect to the FM state for
Rh/Co/Ir(111) films is nearly 200 meV.

The reference system hcp-Co/Re(0001) shows a quite different behavior. The
dispersion profile increases sharply from the $\overline{\Gamma}$ point along
the $\overline{\Gamma\mathrm{KM}}$ and $\overline{\Gamma\mathrm{M}}$
directions and attains a maximum value around $q$$\approx
0.4\times\frac{2\pi}{a}$ for both directions. Then the curve gradually
decreases with $q$ along $\overline{\Gamma\mathrm{M}}$ and obtains a value of
130 meV at the $\overline{\mathrm{M}}$ point. On the other hand, the curve
decreases sharply after $q$$\approx 0.4\times\frac{2\pi}{a}$ along
$\overline{\Gamma\mathrm{KM}}$ and exhibits a local minimum at the
$\overline{\mathrm{K}}$ point. Again it increases sharply with $q$ and obtains
a value of 130 meV at the $\overline{\mathrm{M}}$ point. In contrast to the
other three curves, the maximum of this curve is obtained along the
$\overline{\Gamma\mathrm{M}}$ direction.

To check the validity of the nearest-neighbor effective spin model, we display
the full energy dispersion curve for all four films obtained by only
considering the effective (nearest-neighbor) exchange constant $J_{\rm eff}$
listed in Table 2 (dashed lines in Fig. 7). It is obvious that the effective
model fits the DFT data very well close to the $\overline{\Gamma}$ point where
the variation is parabolic ($q^{2}$) in nature. As the $q$ value increases,
the curves start to deviate from a parabola, indicating a departure from the
effective model and underlining the need to include interactions beyond
nearest neighbors. To get insight into these exchange interactions, we fitted
the full spin spiral dispersion to the Heisenberg model (Eq. 1). The obtained
exchange constants for hcp-Co/Re(0001), hcp-Rh/Co/Re(0001), fcc-Rh/Co/Re(0001)
and fcc-Pd/Co/Re(0001) are given in Table 3.

A comparison of the effective ($J_{\mathrm{eff}}$) and first ($J_{1}$)
exchange interaction constants reveals that $J_{1}$ is slightly larger for the
Rh/Co and Pd/Co bilayer on Re(0001). Since $J_{1}$ is approximately
proportional to the energy difference between the FM and AFM state, this
quantity is underestimated in the effective spin model. The larger value of
$J_{1}$ is balanced by beyond nearest-neighbor exchange constants of negative
sign which favor an antiparallel alignment of spins. Such an exchange
frustration explains the flat energy dispersions in the vicinity of the
$\overline{\Gamma}$ point (Fig. 3(d), Fig. 4(d) and Fig. 5(d)). Taking this
effect into account is also crucial in terms of skyrmion stability since it
has been shown that the energy barrier between a metastable skyrmion and the
FM state is greatly enhanced by exchange frustration [61]. In addition, new
collapse mechanisms of skyrmions such as the Chimera collapse can occur as
shown in Pd/Fe/Ir(111) [29] and in Rh/Co/Ir(111) [62].

##  IV  Conclusion

We have investigated the possibility of complex noncollinear magnetic ground
states in atomic bilayers consisting of a Co and a 4$d$ transition-metal
layers on the Re(0001) surface using DFT. Our study encompasses three 4$d$
transition metals: Rh, Pd and Ru, which give rise to the combinations Co/Rh,
Co/Pd, Co/Ru, Rh/Co, Pd/Co and Ru/Co. We have also considered hcp and fcc
stacking of the topmost layer resulting in twelve ultrathin films. We have
included a Co monolayer in hcp stacking on Re(0001) as a reference system. It
is worth noting that pseudomorphic submonolayer growth of Co has been
experimentally observed on Re(0001).

We have performed total energy calculations for the ferromagnetic and
antiferromagnetic states on their respective relaxed geometries to obtain the
lowest energy collinear spin structure. Further we calculated the energy
dispersion of spin spirals via DFT including spin-orbit coupling within first-
order perturbation theory. We obtained the effective nearest-neighbor exchange
and DMI constants by fitting the spin spiral energy dispersions without and
with SOC, respectively, in the vicinity of the FM state. The MAE constant is
directly computed using DFT based on the energy difference between the in-
plane and out-of-place magnetization directions in the FM state. Our results
reveal that the magnetic interactions strongly depend on the composition and
symmetry of the bilayer due to the hybridization of the Co layer with the $4d$
transition-metal layer and the Re substrate.

Bilayers with a Co overlayer show a notably higher effective exchange constant
than films with a sandwiched Co layer, indicating a steeper rise of the energy
of spin spirals around the FM state for the former set of films. For films
with a Co overlayer, the DMI constant is not large enough to stabilize a spin
spiral state and the FM states remains the ground state. The curvature of the
dispersion curve becomes flatter as the exchange interaction constant reduces
for hcp-Rh/Co/Re(0001) and fcc-Rh/Co/Re(0001), and a higher DMI constant
promotes right-rotating cycloidal spin spirals over the FM state. However, the
MAE counteracts the spin spiral energy in fcc-Rh/Co/Re(0001) and stabilizes a
FM ground state. In hcp-Rh/Co/Re(0001), the MAE can not completely compensate
the spin spiral energy, resulting in a spin spiral ground state in this film
with a period of about 13 nm. The dispersion curve becomes even flatter as the
exchange constant further reduces for hcp-Pd/Co/Re(0001), fcc-Pd/Co/Re(0001)
and fcc-Ru/Co/Re(0001) and an interplay between the DMI and MAE exhibit a FM
ground state.

The dispersion profile of fcc-Rh/Co/Re(0001), hcp-Pd/Co/Re(0001), fcc-
Pd/Co/Re(0001) and fcc-Ru/Co/Re(0001) reveals that these films have potential
to host isolated skyrmions at zero external magnetic field, whereas in case of
hcp-Rh/Co/Re(0001), it necessitates an applied magnetic field of approximately
3 Tesla.

In conclusion, our first-principles DFT calculations show that atomic Rh/Co,
Pd/Co and Ru/Co bilayers on the Re(0001) surface are extremely promising films
in terms of observing magnetic skyrmions at finite and zero applied magnetic
fields. Realizing zero-field magnetic skyrmions in ultrathin films on a
surface which becomes superconducting at low temperature such as Re(0001) is
particularly intriguing. Experimentally, nanoscale magnetic skyrmions have
been realized in ultrathin Pd/Fe films on the Re(0001) surface, however, a
significant external magnetic field is required [45].

##  V  Acknowledgments

We gratefully acknowledge the North-German Supercomputing Alliance for
providing the computing time. S.P. graciously acknowledges the support from
IISER Thiruvananthapuram for funding and computing time at the Padmanabha
cluster. S.P. also acknowledges the computing time from CDAC, India under the
project “topomagnetic-acc”. S.P. further acknowledges financial support from
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (Grant No. 856538, project “3D MAGiC”.

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