††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Thermal evolution of skyrmion formation mechanism in chiral multilayer films

Xiaoye Chen<EMAIL_ADDRESS>Institute of Materials Research &
Engineering, Agency for Science, Technology & Research (A*STAR), 138634
Singapore  Edwin Chue  Physics Department, National University of Singapore
(NUS), 117551 Singapore  Jian Feng Kong  Institute of High Performance
Computing, Agency for Science, Technology & Research (A*STAR), 138632
Singapore  Hui Ru Tan  Institute of Materials Research & Engineering, Agency
for Science, Technology & Research (A*STAR), 138634 Singapore  Hang Khume Tan
Institute of Materials Research & Engineering, Agency for Science, Technology
& Research (A*STAR), 138634 Singapore  Anjan Soumyanarayanan<EMAIL_ADDRESS>Physics Department, National University of Singapore (NUS), 117551 Singapore
Institute of Materials Research & Engineering, Agency for Science, Technology
& Research (A*STAR), 138634 Singapore

###### Abstract

Magnetic skyrmions form in chiral multilayers from the shrinking or fission of
elongated stripe textures. Here we report an experimental and theoretical
study of the temperature dependence of this stripe-to-skyrmion transition in
Co/Pt-based multilayers. Field-reversal magnetometry and Lorentz microscopy
experiments over 100 – 350 K establish the increased efficacy of stripe-to-
skyrmion fission at higher temperatures — driven primarily by the thermal
evolution of key magnetic interactions — thereby enhancing skyrmion density.
Atomistic calculations elucidate that the energy barrier to fission governs
the thermodynamics of the skyrmion formation. Our results establish a
mechanistic picture of the stripe-to-skyrmion transition and advance the use
of thermal knobs for efficient skyrmion generation.

## Introduction

Magnetic skyrmions are topological spin textures arising from the interplay of
collinear and chiral magnetic interactions [1, 2, 3]. They exhibit myriad
technologically desirable characteristics within multilayer thin films such as
ambient stability [4, 5], tunable, nanoscale sizes [6, 7], topological
protection [8, 9, 10], and efficient coupling to electrical currents [11, 12,
13]. These attributes have prompted explorations of their use as building
blocks for next-generation computing [14, 15, 16]. One crucial prerequisite
for most applications is the controlled generation of skyrmions within device
configurations.

Several works have investigated skyrmion nucleation within a uniformly
magnetized background — using constrictions [12, 17], spatial defects [18,
19], or localized heating [20, 21]. Meanwhile, the zero-field (ZF)
configuration for chiral multilayers typically comprises elongated stripes,
which transform into skyrmions at moderate out-of-plane (OP) fields [11, 22,
4, 13, 6]. Recent works have shown that such stripes can also be transformed
into metastable ZF skyrmions by applied electrical currents [23, 24, 25, 26].
While this __driven stripe-to-skyrmion transition is largely attributed to
thermal excitations[23, 25, 26], the associated mechanism and energetics
remain to be established. Moreover, the temperature dependence of this
transition, crucial to its practical utility, remains unaddressed.

Skyrmion formation is governed by magnetic interactions describing domain wall
(DW) energetics [27, 28]. For multilayers hosting the interfacial
Dzyaloshinskii-Moriya interaction (iDMI), chiral domain stability can be
described by the dimensionless parameter $\kappa$ [27, 28, 6], defined as
$\kappa=\pi D/4\sqrt{AK_{\text{eff}}}$, where $D$ is the iDMI, $A$ is the
direct exchange. The effective anisotropy, $K_{\text{eff}}\equiv K_{\rm
u}-\mu_{0}M_{\rm s}^{2}/2$ includes uniaxial ($K_{\rm u}$) and shape
anisotropy ($-\mu_{0}M_{\rm s}^{2}/2$) contributions, where $M_{\rm s}$ is the
saturation magnetization. Recently, it has been shown that for chiral
multilayers with $\kappa>1$, skyrmions may be formed from the break-up, or
“fission” of stripes [29, 30]. Temperature may influence the transition
energetics and kinetics, and therefore merits systematic investigation.

Here, we report on the temperature dependence of the stripe-to-skyrmion
transition using thermodynamic and microscopic probes. Across four chiral
multilayers over 100–350 K, using first-order reversal curve (FORC)
magnetometry, we find that features associated with stripe fission become more
prominent with increasing temperature. Concomitantly, Lorentz transmission
electron microscopy (LTEM) measurements establish that each stripe fissions
into more skyrmions, thereby enhancing the skyrmion density. These
experimentally measured signs of the enhanced propensity of fission with
increasing temperature are correlated with an increase in $\kappa$. With
geodesic nudged elastic band (GNEB) calculations, we show that the observed
temperature evolution of the stripe-to-skyrmion transition is ultimately due
to the reduction in energy barriers to fission, thereby boosting the rate of
this transition.

This work leverages [Ir(10)/Fe($x$)/Co($y$)/Pt(10)]14 multilayer platform
(thickness in angstroms in parentheses), wherein $D$ and $K_{\text{eff}}$ can
be widely varied via the ratio of Fe and Co thicknesses [6]. The four samples
studied here — sputter deposited on Si/SiO2 substrates (see SM1) — have 1 nm
thick FM layers, and are referred to by their Fe($x$)/Co($y$) composition.
They were shown to host sub-100 nm Néel-textured skyrmions whose RT structure
and stability have been studied extensively [29, 30]. Magnetization data were
acquired using an EZ11 vibrating sample magnetometer (VSM) from MicroSense™ in
out-of-plane (OP) geometry. Magnetic imaging used an FEI Titan 80–300 TEM
operated in Fresnel mode at 300 kV, with a defocus of $-2.4$ mm (see SM3),[31]
and sample tilted 15° from normal incidence. The sample temperature was varied
using a Gatan 636 cryogenic holder in automatic or bake-out mode, and OP
fields were applied using the objective lens. Both experiments were performed
over temperatures of 100–350 K, and applied OP magnetic fields ($\mu_{0}H$)
are herein referenced relative to the OP saturation field ($\mu_{0}H_{\rm
s}$).

## FORC Magnetometry

Figure 1: FORC magnetometry and irreversibility. (a, b) Out-of-plane (OP)
magnetization FORCs for samples Fe(0)/Co(10) (a) and Fe(3)/Co(7) (b) at 300 K.
Field was swept from $-H_{\rm s}$ to the reversal field, $H_{\rm r}$
(circles), and back to $-H_{\rm s}$. Arrows show sweep protocol for major
(blue, $H_{\rm r}>H_{\rm s}$) and minor (red, $H_{\rm r}<H_{\rm s}$) loops.
Dotted box is the region of interest for FORC analysis. (c, d) Set of FORCs at
300 K for both samples for selected $H_{\rm r}$ (circles, major loop in
black). Zoomed insets show the $\sim 0.9H_{\rm s}$ convergence-divergence (c)
and $\sim 0.7H_{\rm s}$ divergence (d) of proximate FORCs (green, orange).
(e-f) Color plot of FORC irreversibility, $\rho(H,H_{\rm r})$ (Eq. 1) for both
samples. Irreversible processes, e.g. domain annihilation ($\mathcal{A}$, (e)
inset) and fission ($\mathcal{F}$, (f) inset) present as non-zero $\rho$
features.

A FORC is a field segment of a minor hysteresis loop of magnetization,
$M(H,H_{\rm r})$, characterized by a reversal field $H_{\rm r}$ ($\lesssim
H_{\rm s}$) [32, 33, 34]. The FORC field range for our work (up to $\pm 350$
mT) was determined by acquiring full OP hysteresis loops ($M(H)$) at each
temperature. Subsequently, the samples were negatively saturated
($\mu_{0}H<-1.2~{}\mu_{0}H_{\rm s}$), and the field swept up to $\mu_{0}H_{\rm
r}$, and reversed, and data were acquired till $-10$ mT. A set of such FORCs
were acquired at 2 mT intervals in $\mu_{0}H_{\rm r}$ over the stated
temperatures for all samples. The resulting $M(H,H_{\rm r})$ data were fit to
a second-order polynomial surface (see SM2), and were used to determine the
irreversibility, $\rho(H,H_{\rm r})$ [32]

$\rho(H,H_{\rm r})=-\frac{1}{2}\frac{\partial^{2}M(H,H_{\rm r})}{\partial
H\partial H_{\rm r}}.$ (1)

Figure 2: Thermal Evolution of Irreversibility. (a-d) Zoom-ins of Fe(3)/Co(7)
$M(H)$ loops of at 100, 200, 300, and 350 K respectively. $H_{\rm s}$ (red,
polarized) and $H_{\rm s}^{\prime}$ (black, unpolarized) indicate saturation
fields, and blue arrow indicates their difference ($H_{\rm s}^{\prime}–H_{\rm
s}$). (e-h) Color plots of FORC irreversibility, $\rho(H,H_{\rm r})$ at the
respective temperatures. Red arrow indicates $\Delta H_{FA}$, the separation
between $\mathcal{A}$ and $\mathcal{F}$ features (defined in (h)), dotted blue
arrow indicates ($H_{\rm s}^{\prime}–H_{\rm s}$, as in (a)). (i) Evolution of
$\Delta H_{FA}$, normalised to $H_{\rm s}$ with temperature $T$. Error bars
represents the total peak fit uncertainty (see SM2), the line is a guide-to-
the-eye.

Some magnetization reversal processes are smooth, e.g., spin canting or domain
shrinking/expansion, and are can be reversed with applied field. Others, e.g.
domain switching, are abrupt, and are therefore irreversible. FORC
magnetometry is used to precisely identify and characterize such irreversible
processes, notably via $\rho(H,H_{\rm r})$ — which is nonzero only in
corresponding field regions [32, 33, 34]. FORC had been used extensively study
domain phenomenology in Co/Pt multilayers [33], as illustrated in Fig.
1(a,c,e) for sample Fe(0)/Co(10) at RT. Here, the only discernible
$\rho(H,H_{\rm r})$ feature is a peak-valley pair at $H_{\rm r}\sim
0.9~{}H_{\rm s}$. Labelled as $\mathcal{A}$, it arises from the irreversible
annihilation of domains near saturation (Fig. 1(e), inset), due to sequential
divergence and convergence of proximate FORCs (Fig. 1(c), inset), whose
microscopic origin is well-established [33]. Recently, skyrmion formation in
some samples was found to produce an additional $\rho(H,H_{\rm r})$ feature
[29], shown in Fig. 1(f) for sample Fe(3)/Co(7). For $\kappa\gtrsim 1$
multilayers, individual stripes fission into multiple skyrmions at $H\sim
0.6~{}H_{\rm s}$ (Fig. 1(f), inset). This irreversible stripe-to-skyrmion
fission imprints on $\rho(H,H_{\rm r})$ an additional “$\mathcal{F}$” peak,
and ensues in dense skyrmion configurations and enhanced ZF stability [23, 29,
24]. In the absence of fission, e.g. for $\kappa<1$ multilayers, stripes would
instead shrink into individual skyrmions [30].

Figure 3: LTEM Imaged Texture Density Evolution. (a,c,f,h) LTEM images of
textural field evolution of Fe(3)/Co(7) at 100 K (a), 200 K (c), 300 K (f),
and 350 K (h). Dotted white lines highlight individual stripe-to-skyrmion
transitions. (b,d,e,g) Field dependence of measured skyrmion ($n_{\rm S}$,
black) and stripe densities ($n_{\rm R}$, red). Stars represent maximum
densities ($n_{\rm S}^{\rm max}$ and $n_{\rm R}^{\rm max}$, see (d)). Error
bars represent confidence in texture identification (see SM3). (i) Average
number of skyrmions formed per stripe, $\eta_{\rm RS}\equiv n_{\rm S}^{\rm
max}$ / $n_{\rm R}^{\rm max}$, plotted against temperature. Lines in
(b,d,e,g,i) are guides-to-the-eye.

We now turn to our key finding — the temperature dependence of the stripe-to-
skyrmion transition — by examining its FORC signatures on Fe(3)/Co(7).
Firstly, the $M(H)$ hysteresis loops vary discernibly over 100 – 350 K (Fig.
2(a-d)). The extent of shear — i.e., difference in saturation fields for
polarized ($H_{\rm s}^{\prime}$) and unpolarized ($H_{\rm s}$) curves (Fig.
2(a-d), blue arrows) — is considerable at 100 K, and reduces with increasing
temperature. This corresponds to the $H$-position of the annihilation
($\mathcal{A}$) peak in $\rho(H,H_{\rm r})$ plots (Fig. 2(e-g), blue arrows).
At 100 K, $\mathcal{A}$ is distant from the $H=H_{\rm r}$ diagonal, and
gradually inches towards it with increasing temperature.

More interestingly, the fission ($\mathcal{F}$) peak evolves markedly with
temperature. At 100 K (Fig. 2(e)), $\mathcal{F}$ is nearly merged with
$\mathcal{A}$. With increasing temperature (Fig. 2(f-h)), $\mathcal{F}$ breaks
away from $\mathcal{A}$ and migrates towards the $H=H_{\rm r}$ diagonal. The
separation of $\mathcal{F}$ and $\mathcal{A}$ peaks along $H_{\rm r}$ can be
quantified by $\Delta H_{\mathcal{F}\mathcal{A}}$ (Fig. 2(e), inset, also see
SM2) [29], and is indicative of the prevalence of fission. Indeed, $\Delta
H_{\mathcal{F}\mathcal{A}}/H_{\rm s}$ increases sharply over 200-350 K (Fig.
2(i)), implying that at higher temperatures, fission occurs earlier relative
to annihilation, and skyrmions thus formed are stable over a larger field
range. Overall, the FORC results indicate increased dominance of the fission
process with temperature.

## Lorentz Microscopy

To directly visualize the stripe-to-skyrmion transition, we used LTEM, which
images magnetic textures via Lorentz force deflection of incident electrons
[35, 36]. Samples were negatively saturated ($-H_{\rm s}$), and as the field
was swept to $+H_{\rm s}$, LTEM images were recorded over 8-10 $\mu$m fields-
of-view. Subsequently, non-magnetic contributions were removed by subtracting
a background image acquired above saturation (see SM3). Expectedly, we observe
a labyrinthine configuration at ZF, which transforms, with increasing field,
into stripes, and then into 50-80 nm sized Néel skyrmions (see SM3) [30].
First, we examine the microscopic field evolution of an individual Fe(3)/Co(7)
stripe (Fig. 3(a,c,e,g)). At 100 K (Fig. 3(a)), the highlighted stripe at 0.62
$H_{\rm s}$ shrinks in length as $H$ is increased, eventually into a single
skyrmion at 0.71 $H_{\rm s}$. In contrast, at 300 K (Fig. 3(f)), the
highlighted stripe at 0.55 $H_{\rm s}$ fissions to produce four skyrmions at
0.7 $H_{\rm s}$.

The thermal evolution for individual stripes is consistent with statistical
analyses of textures over $\sim$ 5 – 7 $\mu$m fields-of-view (see SM3). Fig.
3(b,d,e,g) show the field evolution of the densities of stripes ($n_{\rm R}$)
and skyrmions ($n_{\rm S}$) for Fe(3)/Co(7) over 100–350 K. We can quantify
the efficacy of the fission process by $\eta_{\rm RS}(T)$, the average number
of skyrmions formed per stripe. Here, we can approximate $\eta_{\rm RS}$
$\approx n_{\rm S}^{\rm max}$ / $n_{\rm R}^{\rm max}$, where ($n_{\rm S}^{\rm
max}$ and $n_{\rm R}^{\rm max}$ are the maximum skyrmion and stripe densities
at a given temperature (see Fig. 3(d)). As shown in Fig. 3(i), $\eta_{\rm RS}$
displays a monotonic rise with temperature, from $\sim 1$ at 100 K to $\sim$ 3
at 350 K. This suggests that at higher temperatures each stripe fissions into
more skyrmions. Meanwhile, the suppression of fission at lower temperatures
results in the shrinking of stripes to skyrmions. Finally, we note that the
increase in $\eta_{\rm RS}$ expectedly lead to higher $n_{\rm S}^{\rm max}$,
i.e. dense skyrmion configurations [29].

Figure 4: Thermodynamic evolution of stripe-skyrmion transition. (a,b)
Temperature dependence of saturation magnetisation $M_{\rm s}$ (a) and
estimated $\kappa$ (b, see text) for the three studied samples. For (a),
dashed lines indicate vertical offsets (0.05 MA/m each), dotted curves
represent Bloch $T^{3/2}$ fits. (c,d) Compiled variation of $\Delta
H_{\mathcal{F}\mathcal{A}}$ (from FORC $\rho(H,H_{\rm r})$) and $\eta_{\rm
RS}$ ($\equiv n_{\rm S}^{\rm max}/n_{\rm R}^{\rm max}$, from LTEM) with
$\kappa_{\rm est}$ across samples and temperatures. Schematic insets (c-d)
show definitions of $\Delta H_{\mathcal{F}\mathcal{A}}$ (for (c)) and $n_{\rm
S}^{\rm max}$, $n_{\rm R}^{\rm max}$ (for (d)) respectively. Lines in (b-d)
are guides-to-the-eye.

Together, these results on Fe(3)/Co(7) indicate that as temperature increases,
stripe fission becomes the dominant skyrmion formation mechanism. Temperature
variations are known to affect textural transitions in chiral multilayers via
(a) thermal fluctuations [23], and/or (b) magnetic parameter evolution [21].
To delineate these effects, we perform similar experiments over 100 – 350 K on
two more samples — Fe(2)/Co(8) and Fe(4)/Co(6) — providing a wide range of
magnetic parameters [29, 30]. Consistently across samples, the saturation
magnetization $M_{\rm s}$ follows the $T^{3/2}$ Bloch’s law (Fig. 4(a)), and
is expected to to influence the key magnetic interactions as follows:
$A(T)\propto M_{\rm s}(T)^{1.8}$; $D(T)\propto M_{\rm s}(T)^{1.8}$, and
uniaxial anisotropy $K_{\rm u}(T)\propto M_{\rm s}(T)^{2.6}$ (see SM1) [37,
38, 4, 39, 21]. Using these relations, we calculate the expected thermal
evolution of $\kappa$ ($\simeq\kappa_{\rm est}$) across samples. Fig. 4(b)
shows that $\kappa_{\rm est}$ increases monotonically with temperature, with
marked variation on either side of unity ($\sim 0.6-2.8$). We note that for
all three samples, across the entire temperature range of investigation, $D$
is approximately 3 – 6 times greater than the DW anisotropy energy, [40, 41]
which consistently results in the stabilization Néel DWs despite the lower
magnetostatic energy of Bloch DWs.

Fig. 4(c-d) show the compiled evolution of FORC-measured $\Delta
H_{\mathcal{F}\mathcal{A}}/H_{\rm s}$ and LTEM-measured stripe-skyrmion ratio,
$\eta_{\rm RS}$ with $\kappa_{\rm est}$. Consistently across samples and
temperatures, we find that as $\kappa_{\rm est}$ increases from $\sim 0.6$ to
$\sim 2.8$, both $\Delta H_{\mathcal{F}\mathcal{A}}/H_{\rm s}$ ($\sim 0$ to
$\sim 0.3$) and $\eta_{\rm RS}$ ($\sim 1$ to $\sim 3$) show increasing trends.
While $\Delta H_{\mathcal{F}\mathcal{A}}/H_{\rm s}(T)$ and $\eta_{\rm RS}(T)$
also grow monotonically for each sample (see SM4), Fig. 4(c-d) show that the
$\kappa$-dependence of both characteristics collapses onto a single curve
across samples and temperatures. It is likely that magnetic parameters — not
thermal excitations [23] — drive the temperature evolution of the stripe-to-
skyrmion transition, and $\kappa$ plays an important role in the fission
process.

## Atomistic Calculations

Figure 5: Stripe-Skyrmion Transition Energetics. (a) Schematic energy surface,
showing the minimum energy path (red arrows) between two quasi-equilibrium
states (green dots). White dots show intermediate states. (b) GNEB simulated
magnetization images of stripe-to-skyrmion fission for Fe(3)/Co(7) parameters
at $T=200$ K, showing initial (1), intermediate (2-3), and final (4)
magnetization states. (c) Energy profile for the transition in (b), with
positions of states (1-4) overlaid. The energy barrier, $E_{\rm b}$ (red
arrow) and net energy difference, $\Delta E$ (blue arrow) – are indicated.
(d-f) Plots of $\Delta E(T)$ (d), $E_{\rm b}(T)$ (e), and transition lifetime
($\tau_{\rm RS}(T)$ ((f), from Eq. 2) against temperature for stripe-to-
skyrmion fission for Fe(3)/Co(7) parameters. Solid lines in (d-f) are guides-
to-the-eye.

The fission of a stripe into multiple skyrmions must overcome topological
protection and the associated energy barrier [42]. To understand the fission
energetics, we turn to GNEB calculations performed using the Fidimag package
[43]. GNEB calculates the minimum energy path for transitions between two
fixed quasi-equilibrium magnetic states (schematic: Fig. 5(a)), while
implicitly constraining magnetic moments to fixed magnitude [44, 45, 46]. The
energy, $E$ of the magnetic configurations is calculated using the atomistic
Heisenberg Hamiltonian and referenced to the initial state [47].

Here the initial state was chosen to be a stripe (Fig. 5(b): (1)) and the
final state to be two-skyrmions (Fig. 5(b): (4)), and these states were first
relaxed using the atomistic Landau-Lifshitz-Gilbert equation under a 50 mT
magnetic field [46, 43]. Subsequently GNEB was used to determine the
transition energy profile (shown in Fig. 5(c)). Temperature dependence was
incorporated using rescaled magnetic interactions (scaling laws in Fig. 4).
First, Fig. 5(d) shows that the transition energy difference, $\Delta E<0$ at
all temperatures, i.e. the two-skyrmion state is consistently energetically
favoured to the stripe state. While $\Delta E(T)$ does show up to 20$\%$
variation, the trend is non-monotonic. In contrast, the transition energy
barrier $E_{\rm b}(T)$ (Fig. 5(e)) exhibits a sharp ($\sim 3\times$),
monotonic reduction with increasing temperature. Thus, we posit the direct
association of $E_{\rm b}(T)$ with the observed increase in stripe-to-skyrmion
fission activity.

We estimate the stripe-to-skyrmion transition lifetime, $\tau_{\rm RS}$, using
Néel-Arrhenius relaxation theory as [47]

$\tau_{\rm RS}=\tau_{0}\exp\left(E_{\rm b}/k_{\rm B}T\right),$ (2)

where $\tau_{0}$ is the attempt period (typically $\sim 10^{-9}--10^{-12}$ s
[47, 48, 49]) and $k_{\rm B}$ is the Boltzmann constant. With $\tau_{0}$
$\sim$ 10-12 s, we find that $\tau_{\rm RS}$ reduces dramatically from $\gg
10^{50}$ s at 100 K to $\sim 0.1$ s at 300 K. This order-of-magnitude
transition rate estimate supports the hypothesis that over 100–350 K, stripe-
to-skyrmion fission for Fe(3)/Co(7) evolves from kinematically forbidden to
rapidly accessible. In contrast, if we assume a temperature-independent energy
barrier fixed at its 100 K value, the transition lifetime remains extremely
long at $\tau_{RS}>10^{10}~{}s$ at 300 K, which contradicts our experimental
observations. This confirms that thermal fluctuations alone cannot account for
the observed $T$-dependence of fission and that $\kappa(T)$-driven energy
barrier reduction is crucial to enabling fission kinematics, which, in turn,
manifest in the temperature dependence of $\Delta H_{\mathcal{F}\mathcal{A}}$
and $\eta_{\rm RS}$.

## Summary

In summary, we have presented detailed thermodynamic and microscopic evidence
on the temperature evolution of stripe-to-skyrmion fission in chiral
multilayers. The increased prominence of the magnetometry fission peak and
enhanced fission efficacy in LTEM are both well-described by the thermal
evolution of the DW stability parameter $\kappa$ across samples. Atomistic
calculations establish that this arises from the sharp reduction of the energy
barrier to stripe fission at elevated temperatures. Our work provides timely
insights on the thermodynamics of the stripe-to-skyrmion transition. On one
hand, while anisotropy and temperature are known to independently enhance
skyrmion density [6, 50, 29, 51], these results establish a much-needed bridge
between the thermodynamics and microscopics. On the other hand, these
thermodynamic effects can be exploited, e.g. via controlled temperature
cycles, to generate skyrmion textures [23, 25, 26, 24, 21]. Our framework may
be particularly relevant to manipulating chiral spin texture ensembles for
unconventional computing [16].

We acknowledge the support of the National Supercomputing Centre (NSCC),
Singapore, for computational resources. This work was supported by the SpOT-
LITE program (Grant No. A18A6b0057), funded by Singapore’s RIE2020
initiatives. E.C. was partially supported by the NUS Resilience and Growth
Traineeship Programme.

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Supplementary Materials

## S1. Magnetic Properties and Temperature Dependence

The Ta(40)/Pt(50)/[Ir(10)/Fe($x$)/Co($y$)/Pt(10)]14/Pt(20) multilayer samples
used in this work (layer thickness in angstroms in parentheses) and their room
temperature (RT) magnetic properties are tabulated in Table S1. The RT
magnetic properties of multilayer samples with identical compositions
sputtered using similar deposition parameters have been extensively
characterized in our previous works [29, 30]. Here, the RT saturation
magnetisation, $M_{\rm s}$, and effective anisotropy, $K_{\rm eff}$, were
determined from VSM measurements. Meanwhile, the RT values of exchange
stiffness ($A_{\rm est}$) and interfacial Dzyaloshinskii-Moriya interaction
iDMI ($D_{\rm est}$) were derived from our previous works, which used
identical stack compositions [29, 30].

Acronym | Stack Composition | $\boldsymbol{M_{\rm s}}$ | $\boldsymbol{K_{\rm eff}}$ | $\boldsymbol{D_{\rm est}}$ | $\boldsymbol{A_{\rm est}}$ | $\boldsymbol{\kappa}$
---|---|---|---|---|---|---
(MA/m) |  (MJ/m3) |  (mJ/m2) | (pJ/m)
Fe(0)/Co(10) |  [Ir(10)/Co(10)/Pt(10)]14 | 1.16 | 0.60 | 1.2 | 17.8 | 0.3
Fe(2)/Co(8) |  [Ir(10)/Fe(2)/Co(8)/Pt(10)]14 | 1.14 | 0.26 | 0.9 | 12.8 | 0.4
Fe(3)/Co(7) |  Ir(10)/Fe(3)/Co(7)/Pt(10)]14 | 1.11 | 0.095 | 1.7 | 13.2 | 1.2
Fe(4)/Co(6) |  [Ir(10)/Fe(4)/Co(6)/Pt(10)]14 | 1.05 | 0.058 | 1.9 | 13.6 | 1.6
Table S1: Sample Compositions and RT Magnetic Properties. List of samples used
in this work, with Fe($x$)/Co($y$) acronyms, active stack compositions (layer
thickness in angstroms in parentheses), and key RT magnetic properties —
$M_{\rm s}$, $K_{\rm eff}$, $D_{\rm est}$ , $A_{\rm est}$, and $\kappa$. Note:
Fe(0)/Co(10) is used only for illustrative purposes (manuscript Fig.
1(a,c,e)). Figure S1: Measured $T$-dependence of Fe(3)/Co(7) magnetic
parameters. (a) Out-of-plane $M(H)$ measurements over $H>H_{\rm s}$, performed
for 10 K to 320 K. Lines show linear fits, whose $y$-intercepts correspond to
$M_{\rm s}$. (b) The $T$-dependence of $M_{\rm s}$, as determined from (a).
Line shows a Bloch law ($T^{3/2}$) fit. (c) Logarithmic scatter plot of the
measured uniaxial anisotropy, $K_{\rm u}=K_{\rm eff}+\mu_{0}M_{\rm s}^{2}/2$
against $M_{\rm s}$ across temperatures. Line shows a linear fit with slope
$\approx 2.6$.

To determine the temperature ($T$)-dependence of magnetic parameters ($A_{\rm
est}$, $D_{\rm est}$) for manuscript Fig. 4, we first measured the
$T$-dependence of $M_{\rm s}$ using an MPMS-SQUID magnetometer from Quantum
Design™. To correctly account for the $T$-dependent diamagnetic contribution
of the substrate, we performed at each temperature a magnetic field sweep at
fields $H$ greater than the saturation field, $H_{\rm s}$, of the sample . As
shown in Figure S1(a) for sample Fe(3)/Co(7), each field sweep thus acquired
was fit to a straight line, and the $M_{\rm s}$ was read off as the
$y$-intercept. Subsequently, as shown in Figure S1(b), the $M_{\rm s}(T)$ data
was fit to the Bloch $T^{3/2}$ law, $M_{\rm s}(T)=M_{0}(1-\beta T^{3/2})$.
$M_{\rm s}(T)$ was then used to estimate the $T$-dependence of other magnetic
parameters ($A_{\rm est}$, $D_{\rm est}$) using established scaling relations
detailed in the manuscript.

Finally, the $T$-dependence of anisotropy was accounted for by performing in-
plane and out-of-plane $M(H)$ measurements over the temperature range of
interest. Figure S1(c) shows a plot of the empirically measured scaling
relationship of uniaxial anisotropy $K_{\rm u}$ with $M_{\rm s}$. The observed
relationship — $K_{\rm u}\propto M_{\rm s}^{2.6}$ — is similar to previous
works which estimated $K_{\rm u}\propto M_{s}^{3}$ [37, 4, 38].

## S2. FORC Magnetometry Measurements and Analysis

Figure S2: VSM and FORC measurements. (a). Set of FORCs obtained for sample
Fe(3)Co(7), with $H_{r}$ and H at intervals of 2 mT. (b). Color plot of
magnetisation for the FORC data in (a), projected on H, $H_{r}$ axis. (c).
Color plot of $\rho$, the FORC irreversibility, derived from the data in
(a-b).

The color plots of FORC irreversibility, $\rho(H,H_{\rm r})$ — shown in
manuscript Fig. 1(e-f), and reproduced in Figure S2(c) — were obtained by
processing the set of VSM-measured FORCs, shown e.g. in Figure S2(a). First,
the measured magnetization, $M(H,H_{\rm r})$ (Figure S2(b)), was fit to a
second-order polynomial surface [32, 34]:

$M(H,H_{\rm r})=a_{1}+a_{2}H_{\rm r}+a_{3}H+a_{4}H_{\rm
r}^{2}+a_{5}H^{2}+a_{6}H_{\rm r}H.$ (S1)

The number of points used for fitting was determined by a smoothing factor,
which is 5 in our work. Following from Equation S1, $\rho(H,H_{\rm r})$,
defined as

$\rho(H,H_{\rm r})=-\frac{1}{2}\frac{\partial^{2}M(H,H_{\rm r})}{\partial
H\partial H_{\rm r}},$ (S2)

is given by $-a_{6}$.

To quantify the separation between the prominent irreversible features —
$\mathcal{A}$ and $\mathcal{F}$ — in FORC $\rho(H,H_{\rm r})$ plots ($\Delta
H_{\mathcal{F}\mathcal{A}}$, see manuscript Fig. 2), it is necessary to
identify the centres of the respective features. This was accomplished using
an algorithm written in Python code, as follows. First, the FORC distribution
was binarized to identify $\rho>0$ regions. Next, a watershed segmentation
procedure was implemented for FORC distributions that visually exhibited
overlap of two distinct features. The centres of these features in ($H,H_{\rm
r}$) space were then identified by appropriately taking the weighted averages,
thus enabling the determination of $\Delta H_{\mathcal{F}\mathcal{A}}$.
Meanwhile, the standard deviation thus obtained indicates the spread of these
features in ($H,H_{\rm r}$) space.

## S3. Lorentz TEM Measurements and Analysis

Figure S3: Background Removal for LTEM Images. (a-b) As-acquired LTEM image at
$\mu_{0}H\simeq-220$ mT, depicting a sparse skyrmion configuration (a), and
corresponding “background” (saturated) image at $\mu_{0}H\simeq-310$ mT (b)
for Fe(3)/Co(7) at RT. (c) Processed image for $\mu_{0}H\simeq-220$ mT,
obtained by subtracting (b) from (a) after translating and rotating (a) to
align with (b), with visibly enhanced magnetic texture contrast c.f. (a).

Phase shift $\gamma(k)$ for a scattered beam is caused by spherical aberration
and defocus, where $\gamma(k)=2\pi(C_{s}\lambda k^{4}/4+\Delta f\lambda
k^{2}/2)$ [31]. Dark rings are visible in the diffractogram when the phase
shift is a multiple of $\pi$, i.e. $\gamma(k)=n\pi$. The defocus values
reported in this work were determined from the intercept of a plot of
$n/k^{2}$ against $k^{2}$ from a calibrated diffractogram taken at a
particular defocus.

To enhance the magnetic contrast in Lorentz TEM images and improve the
reliability of the statistical analysis of magnetic textures, we first perform
a background subtraction procedure for each LTEM image to remove prominent
sources of non-magnetic contrast. The procedure utilizes a reference, or
“background” image – without magnetic textures (i.e. captured at saturation) –
over approximately the same field-of-view. This allows the removal of contrast
from non-magnetic sources, such as due to structural inhomogeneity of the
film, waviness of the substrate etc. This process is accomplished, using a
custom-written Python code, by first aligning the two images to correct for
translational and rotational drifts. Next, simple pixel-wise subtraction is
performed. The importance of the background subtraction process is illustrated
in Figure S3. The contrast of magnetic textures is much more prominent in the
processed image (Figure S3(c)) c.f. the original image (Figure S3(a)).

Figure S4: Identifying & Counting Magnetic Textures. Representative processed
LTEM image of Fe(3)/Co(7) acquired at RT, $\mu_{0}H\simeq-240$ mT, with a
mixed configuration with stripes and skyrmions over a $\sim 30$ $\mu{\rm
m}^{2}$ area. Textures identified as skyrmions and stripes are annotated with
circles and crosses respectively. For skyrmions (stripes), blue and green (red
and white) colors indicate certain and uncertain identifications of the
respective magnetic textures.

To quantify the field and temperature evolution of magnetic texture densities
(see e.g. manuscript Fig. 3), every domain observed in LTEM images recorded
over $H\sim(0.5--1)\cdot H_{\rm s}$ (, Figure S5) was visually identified as a
skyrmion or a stripe. An example of the outcome of such texture identification
is shown in Figure S4. To reflect our confidence in the identification of
skyrmions and stripes within dense domain backgrounds, the final skyrmion and
stripe counts are obtained from weighted sums. Specifically, domains that are
certain skyrmions or stripes (Figure S4: blue circles, red crosses) contribute
their full value of unity to the respective domain counts. Meanwhile, domains
of uncertain identity (Figure S4: green circles, white crosses) contribute
half their value to their respective counts, while the other half of their
magnitude is added to the corresponding error bars.

Figure S5: Field Evolution of LTEM-Imaged Textures $M(H)$ curves of
Fe(3)/Co(7), showing field-polarized (red) and unpolarized (black) sweeps
measured by VSM at 100 K (a) and 300 K (b) respectively. Insets show series of
LTEM images recorded along the unpolarized (black) curves at fields
corresponding to dashed black lines.

The LTEM images shown in manuscript Fig. 3 focus on elucidating the
microscopics of the stripe-to-skyrmion transition. In Figure S5, we present
representative LTEM data acquired on Fe(3)/Co(7) at 100 K and 300 K over a
larger field range. In both cases, at zero field, we observe a labyrinthine
stripe configuration, which transforms, with increasing field, first into
stripes, and eventually into skyrmions. Notably, the density of magnetic
textures, esp. skyrmions, is much higher for the 300 K data (Figure S5(b))
compared to that at 100 K (Figure S5(a)).

## S4. Sample Dependence Analysis

Figure S6: Sample-Wise $T$-Dependence of Stripe-to-Skyrmion Transition
Characteristics. Compiled $T$-dependence across the three samples of (a) FORC-
measured separation of $\mathcal{F}$ and $\mathcal{A}$ peaks, $\Delta
H_{\mathcal{F}\mathcal{A}}$, and (b) LTEM-measured ratio of maximal skyrmion
to stripe densities, $\eta_{\rm RS}\equiv n_{\rm S}^{\rm max}$ / $n_{\rm
R}^{\rm max}$. The trends exhibit visible sample-wise variation c.f.
manuscript Fig. 4(c-d).

In manuscript Fig. 4, we examined the thermodynamic variation of both the
measured stripe-skyrmion transition characteristics across samples and
temperatures — (1) the separation, $\Delta H_{\mathcal{F}\mathcal{A}}$, of the
$\mathcal{F}$ and $\mathcal{A}$ peaks measured by FORC magnetometry, and (2)
the ratio of f maximal skyrmion to stripe densities, $\eta_{\rm RS}\equiv
n_{\rm S}^{\rm max}$ / $n_{\rm R}^{\rm max}$. While manuscript Fig. 4(c-d)
showed the variation of $\Delta H_{\mathcal{F}\mathcal{A}}$ and $\eta_{\rm
RS}$ with $\kappa_{est}$ (obtained from manuscript Fig. 4(b)), we present in
Figure S6 the $T$-dependence of both these characteristics. Notably, we find
that while both $\Delta H_{\mathcal{F}\mathcal{A}}$ (Figure S6(a)) and
$\eta_{\rm RS}$ (Figure S6(b)) rise monotonically with $T$ for all three
samples, the trends exhibit visible sample dependence. In contrast, manuscript
Fig. 4(c-d) consistently show the collapse of sample- and $T$-dependent data
onto a single curve. This provides further experimental support for the
crucial role played by the thermodynamic parameter $\kappa$ in determining the
nature of the stripe-to-skyrmion transition.