# Targeted Writing and Deleting of Magnetic Skyrmions
in Two-Terminal Nanowire Devices

Soong-Guen Je  Center for X-ray Optics, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Department of Emerging Materials Science,
Daegu Gyeongbuk Institute of Science and Technology (DGIST), Daegu 42988,
Korea  Department of Physics, Chonnam National University, Gwangju 61186,
Korea  Dickson Thian  Institute of Materials Research & Engineering, Agency
for Science, Technology & Research (A*STAR), 138634 Singapore  Xiaoye Chen
Institute of Materials Research & Engineering, Agency for Science, Technology
& Research (A*STAR), 138634 Singapore  Data Storage Institute, Agency for
Science, Technology & Research (A*STAR), 138634 Singapore  Lisen Huang
Institute of Materials Research & Engineering, Agency for Science, Technology
& Research (A*STAR), 138634 Singapore  Data Storage Institute, Agency for
Science, Technology & Research (A*STAR), 138634 Singapore  Dae-Han Jung
School of Materials Science and Engineering, Ulsan National Institute of
Science and Technology, Ulsan 44919, Korea  Weilun Chao  Center for X-ray
Optics, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Ki-Suk Lee  School of Materials Science and Engineering, Ulsan National
Institute of Science and Technology, Ulsan 44919, Korea  Jung-Il Hong
Department of Emerging Materials Science, Daegu Gyeongbuk Institute of Science
and Technology (DGIST), Daegu 42988, Korea  Anjan Soumyanarayanan
<EMAIL_ADDRESS>Institute of Materials Research & Engineering, Agency
for Science, Technology & Research (A*STAR), 138634 Singapore  Data Storage
Institute, Agency for Science, Technology & Research (A*STAR), 138634
Singapore  Physics Department, National University of Singapore (NUS), 117551
Singapore  Mi-Young Im<EMAIL_ADDRESS>Center for X-ray Optics, Lawrence Berkeley
National Laboratory, Berkeley, California 94720, USA  Department of Emerging
Materials Science, Daegu Gyeongbuk Institute of Science and Technology
(DGIST), Daegu 42988, Korea

###### Abstract

Controllable writing and deleting of nanoscale magnetic skyrmions are key
requirements for their use as information carriers for next-generation memory
and computing technologies. While several schemes have been proposed, they
require complex fabrication techniques or precisely tailored electrical
inputs, which limits their long-term scalability. Here we demonstrate an
alternative approach for writing and deleting skyrmions using conventional
electrical pulses within a simple, two-terminal wire geometry. X-ray
microscopy experiments and micromagnetic simulations establish the observed
skyrmion creation and annihilation as arising from Joule heating and Oersted
field effects of the current pulses, respectively. The unique characteristics
of these writing and deleting schemes, such as spatial and temporal
selectivity, together with the simplicity of the 2-terminal device
architecture, provide a flexible and scalable route to the viable applications
of skyrmions.

## Introduction

Magnetic skyrmions are spatially localized spin textures with a well-defined
topology 1, 2. Their nanoscale size and ambient stability in metallic thin
films 3, 4, 5, 6, as well as efficient coupling to electrical currents 7, 8,
are desirable inherent attributes that have prompted much excitement 9.
Notably, numerous recent device proposals seek to harness the skyrmion motion
within a wire geometry 10 to realize applications in memory, logic, and
unconventional computing 11, 12, 13. An indispensable requirement for such
devices is a spatially and temporally controlled scheme to write and delete
skyrmions through robust and scalable techniques.

There has been considerable progress in writing magnetic skyrmions.
Theoretical studies suggested that skyrmions can be created by spin currents
14, 15, 16, 17. Recent device-level write efforts have mostly utilized spin-
orbit torques (SOTs) in application-relevant materials 9, 18, 19, 17. To
facilitate the SOT-writing, the schemes require additional components, for
example, geometric constrictions 7, 20, 21, and defects 22, 23. While
effective, however, these additional requirements lead to fabrication
complexities in some cases, limiting their scalability.

On the contrary, the electrical deletion of skyrmions has thus far been
lacking, and has been recognized as a critical pending challenge toward
functional skyrmionic devices 22, 23. While two recent schemes have been
reported for deterministic deletion and creation of skyrmions 21, 23, they
have additional prerequisites, such as in-plane bias field 23 or precisely
positioned nanostructures 21. In the other, completely different current paths
are required for writing and deletion 21, increasing the complexity of device
operation. Therefore, the need for the hour is to realize simple schemes for
electrical skyrmion writing and deletion that may be readily adapted to myriad
syrmion applications in a controlled fashion.

Here we present a simple and facile approach for writing and deleting
skyrmions wherein an identical conventional current pulse is used for both
operations. Crucial to this achievement is the harnessing of two by-products
of electrical currents: Joule heating and Oersted field. By exploiting
contrasting spatial and temporal properties of these two effects, we can
achieve targeted operation, e.g., by simply reversing the pulse polarity. The
ubiquitous character of these current-induced effects across material systems,
coupled with minimalist requirements, offers much-needed attributes of
scalability and broad applicability.

Figure 1: Experimental setup and current-driven skyrmion writing experiments.
(a) Schematic of the multilayer thin film used in this work (layer thickness
in nm in parentheses). (b) Schematic of the MTXM experimental setup. (c)
Hysteresis loop for OP magnetization, $M\left(\mu_{0}H\right)$, with
corresponding MTXM images (scale bar: 1 $\mu$m) at OP fields indicated by
vertical dashed lines. Top three images (right to left) show the formation of
chiral magnetic textures as $\mu_{0}H$ is reduced from positive saturation
($+\mu_{0}H_{{\rm s}}$), while the bottom three (bottom to top) show their
shrinking and disappearance as $\mu_{0}H$ is increased from zero to
$+\mu_{0}H_{{\rm s}}$. The shaded grey field region, corresponding to a
uniformly magnetized state, indicates the base field range for experiments in
(d-e) and Fig. 3. (d) Typical examples of MTXM images (scale bar: 1 $\mu$m) of
the wire before and after the current pulse. Top image shows the uniformly
magnetized initial state at $\mu_{0}H_{{\rm b}}$. Bottom images show the final
states (i.e., after a current pulse) at each $\mu_{0}H_{{\rm b}}$. (e) Plot of
the density, $n_{{\rm c}}$, of skyrmions nucleated as per (d) against the base
field $\mu_{0}H_{{\rm b}}$. The inferred crossover field (see Fig. 3 and
associated text) is labelled as $\mu_{0}H_{{\rm c}}$.

This work was performed on multilayer stacks of [Pt(3)/Co(0.9)/MgO(1.5)]15
(Fig. 1a, hereafter Pt/Co/MgO), which were sputtered on X-ray transparent
Si3N4 membranes, and fabricated into 2 $\mu$m wide wires (see Methods). The
asymmetric Pt/Co/MgO stack hosts a sizeable interfacial Dzyaloshinskii-Moriya
interaction (DMI), which can stabilize Néel-textured skyrmions at room
temperature (RT) 4, 24. To observe sub-100 nm magnetic skyrmions in Pt/Co/MgO
multilayers, we employed full field magnetic transmission soft x-ray
microscopy (MTXM) at the Advanced Light Source (XM-1, BL6.1.2) 25. Out-of-
plane (OP) geometry (Fig. 1b) is employed to detect the OP magnetization
through X-ray magnetic circular dichroism. The imaging results, acquired at RT
with in situ OP magnetic field $\mu_{0}H$, are complemented by micromagnetic
simulations using magnetic parameters determined from magnetometry
measurements (see Methods).

We begin by examining the equilibrium magnetization $M(\mu_{0}H)$ of the
Pt/Co/MgO multilayer, and corresponding MTXM images obtained over a wide wire
pad (Fig. 1c). The sheared hysteresis is characteristic of a multi-domain
configuration at remanence as confirmed by the labyrinthine zero field (ZF)
MTXM image. As $\mu_{0}H$ is swept down from positive saturation (orange
branch in Fig. 1c), stripe domains nucleate at $\sim 70$ mT, which propagate
to form a labyrinthine state. Meanwhile, on the upsweep (green branch in Fig.
1c), increasing $\mu_{0}H$ beyond zero results in the labyrinthine state
transforming to oppositely magnetized stripes, which eventually shrink to a
sparse array of sub-100 nm skyrmions. For the subsequent electrical writing
experiments, the used magnetic fields correspond to the shaded region of Fig.
1c, wherein the initial MTXM images are uniformly magnetized. Slight
discrepancies between device level imaging and film-level $M(\mu_{0}H)$
results may be attributed to the difference in the numbers of nucleation sites
between the patterned wire26 and the film 3.

## Electrical Writing of Skyrmions

Figure 2: Simulations of heat-induced skyrmion nucleation. (a) Measured
variation of saturation magnetization ($M_{{\rm S}}$) with temperature after
normalizing to its 300 K value. This data is used along with scaling relations
of magnetic parameters as inputs for Joule heating simulations. (b) Schematic
recipe used for micromagnetic simulations of Joule heating effects. (c)
Typical examples of simulation results for _$\Delta T$_ = 111 K, 165 K and
$\mu_{0}H_{{\rm b}}$ = 90 mT, 120 mT, respectively (scale bar: 0.5 $\mu$m).
Simulations are initialized at $\mu_{0}H>\mu_{0}H_{{\rm s}}$, and the uniform
state is brought to $\mu_{0}H_{{\rm b}}$ (c, top). During the heating phase
(b, centre), the simulated temperature is raised by $\Delta T$ and magnetic
parameters are rescaled (see text, Fig. 4a). Skyrmions are nucleated in some
cases (c, right), and persist when RT conditions are restored. (d) Color plot
showing the nucleated skyrmion density, $n_{{\rm c}}$, as a function of
$\Delta T$ and $\mu_{0}H_{{\rm b}}$ (the latter c.f. Fig. 1e). (e)
Micromagnetic energy difference, $\Delta E$, between final (skyrmion
nucleated) and initial (uniform) states for various _$\Delta T$_ and
$\mu_{0}H_{{\rm b}}$. The final state is more stable if $\Delta E<0$ (note:
_$\Delta E=0$_ for $\Delta T=111$ K as the initial and final states are
identical).

For electrical writing experiments, we begin by saturating the sample at large
positive field, and lower it to a base field $\mu_{0}H_{{\rm b}}$ ranging from
80 to 126 mT (Fig. 1c, shaded grey region) wherein the sample retains uniform
magnetization (Fig. 1d: top). Under these conditions, the injection of a
single current pulse (width $\sim 30$ ns, amplitude $\sim 6.8\times 10^{11}$
A/m2) results in the nucleation of skyrmions in the wire device (Fig. 1d:
bottom). We observe that the number of nucleated skyrmions varies considerably
with $\mu_{0}H_{{\rm b}}$ as seen across the MTXM images (Fig. 1d: bottom).
The skyrmion nucleation is quantified in Fig. 1e by measuring the density of
created skyrmions, $n_{{\rm c}}$, with respect to $\mu_{0}H_{{\rm b}}$. Below
$\mu_{0}H_{{\rm b}}\sim 100$ mT, the current pulse nucleates a dense skyrmion
array, with density $n_{{\rm c}}>$ 5 $\mu$m-2 (e.g.. left in Fig. 1d). Above
$\mu_{0}H_{{\rm b}}\sim$100 mT, $n_{{\rm c}}$ drops sharply with increasing
$\mu_{0}H_{{\rm b}}$ (middle in Fig. 1d), and reaches zero at ~120 mT.
Notably, the modulation of the electrical nucleation of skyrmions by the
external magnetic field is distinct from those reported in previous works 7,
21, 22, 23, 27.

Lateral currents within an asymmetric stack (e.g., Pt/Co/MgO) may produce a
large SOT 18. Such SOT governs skyrmion motion and, in some recent works,
induces skyrmion nucleation 21, 22, 23. However, the SOT-driven writing
mechanisms cannot explain our results for the following reasons. First, our
device does not have any geometric constrictions, positioned defects, or in-
plane magnetic fields required for the SOT writing of skyrmions 21, 22.
Second, unlike the peculiar bipolar pulse waveform used to achieve the dynamic
SOT nucleation 23, we use a conventional single pulse with a rectangular
profile. Crucially, as we will show in Fig. 3, the same pulse can be used for
deletion of skyrmions which clearly rules out vectorial mechanisms associated
with SOT. On the other hand, a plausible explanation may be the Joule heating
effect and associated thermodynamics, which were previously shown to drive
transitions from stripe domains to skyrmions 27. For our case, simulation and
analytical modelling suggest that current pulses used may induce a temperature
rise $\Delta T\sim$ 220 K28.

To investigate the plausibility of heat-induced skyrmion nucleation, we
performed micromagnetic simulations (see Methods). Typically, such simulations
account for temperature effects solely by introducing randomly fluctuating
fields 29. However the $\Delta T$ would additionally change the magnetic
parameters. To consider this effect, we measured the saturation magnetization
$M_{{\rm S}}(T)$ over 300-650 K (Fig. 2a), and estimated the micromagnetic
parameters at elevated temperatures using scaling relations with magnetization
30, 31, 32: exchange stiffness $A\propto M_{{\rm S}}^{1.8}$, interfacial DMI
$D\propto M_{{\rm S}}^{1.8}$ and uniaxial anisotropy $K_{{\rm u}}\propto
M_{{\rm S}}^{3.03}$. The micromagnetic simulations were then performed
following the recipe depicted in Fig. 2b (details in Supporting Information
(SI)). In particular, SOT effects were not included. The system is first
initialized to a uniformly magnetized state at $\mu_{0}H_{{\rm b}}$ (Fig. 2c:
top), subjected to a heating phase with rescaled parameters, and then restored
to RT conditions (Fig. 2c: bottom).

Figure 3: Current-driven skyrmion deletion experiments. (a) Recipe used for
skyrmion deletion, with representative MTXM images (scale bar: 1 µm) as
insets. (b) Plot of the skyrmion annihilation rate, $R_{{\rm a}}\left(\equiv
1-N_{\text{aft}}/N_{\text{bef}}\right)$, with varying base field
$\mu_{0}H_{{\rm b}}$. The inferred crossover field, $\mu_{0}H_{{\rm c}}$, is
indicated (c.f. Fig. 1e, see caption). (c) Plot of the expected Oersted field
profile (inset shows schematic) generated by a current applied along
$-\widehat{y}$ (red arrow) as a function of the transverse position $x$ across
the wire.

Fig. 2c shows the simulation results for representative values of $\Delta T$
(111, 165 K) and $\mu_{0}H_{{\rm b}}$ (90, 120 mT). For $\Delta T$ = 165 K
(Fig. 2c, right), skyrmions are nucleated by the heat pulse, and their number
decreases as $\mu_{0}H_{{\rm b}}$ increases. In contrast, the smaller heat
pulse $\Delta T$ = 111 K (Fig. 2c, left) does not create any skyrmions
regardless of $\mu_{0}H_{{\rm b}}$. Fig. 2d shows in further detail the
simulated $n_{c}$ with $\Delta T$ (100-250 K) and $\mu_{0}H_{{\rm b}}$ (80-120
mT). We find that a threshold $\Delta T>$ 150 K is required to nucleate any
skyrmions, corresponding to a scaled magnetization $M_{s}(300+\Delta T)\sim
0.9M_{s}(300\,{\rm K})$. Above this threshold, the field dependence of
$n_{{\rm c}}$ is qualitatively consistent with the experimental data (Fig.
1e). Overall, this indicates that Joule heating induced by current pulses is a
viable explanation of the observed nucleation of skyrmions and the dependence
of $n_{{\rm c}}$ on $\mu_{0}H_{{\rm b}}$.

Micromagnetic simulations further help to elucidate several aspects of heat-
induced skyrmion energetics. First, Fig. 2e shows the energy difference,
$\Delta E$, between the final and initial (ambient) states with varying
$\Delta T$ and $\mu_{0}H_{{\rm b}}$. Notably $\Delta E$ is negative for most
cases, with negligible variation over $\Delta T$ of 165-228 K, i.e., the
skyrmion formation reduces the total energy with respect to that of the
uniform state. This indicates that skyrmion nucleation is thermodynamically
favoured (explored further in Conclusions), except near saturation
($\mu_{0}H_{{\rm b}}=$ 120 mT) wherein skyrmions may be metastable. Next, we
note a slight qualitative difference in the nucleation phenomenology with
$\Delta T$ (details in SI). For intermediate $\Delta T$ ($<220$ K), skyrmions
are formed during the heating phase. However, for higher $\Delta T$ ($>220$
K), i.e., $M_{s}(T)<0.8M_{s}(300\,{\rm K})$, fluctuations during the heat
phase result in random magnetization, and skyrmions are formed only when the
heat is turned off. Finally, simulations allow for a sequential study of the
two heat-induced effects: thermal fluctuations and rescaling of magnetic
parameters. Crucially, we find that no skyrmions are nucleated at any
$\mu_{0}H_{{\rm b}}$ or $\Delta T$ if magnetic parameters are not rescaled
(see SI). Together, these simulation insights strongly point to the
thermodynamic origin of the observed nucleation of skyrmions. Finally, while
the simulation results in Fig. 2 are grain-free, potential effects of pinning
sites on heating and skyrmion nucleation are discussed in the SI.

## Electrical Deletion of Magnetic Skyrmions

Figure 4: Simulations of Oersted field-induced skyrmion annihilation. (a)
Schematic recipe used for simulating Oersted field effect with Joule heating
(details in SI). The current pulse is modelled as inducing an Oersted field
(profile in Fig. 3c) for the entire 30 ns duration (bottom) and Joule heating
for the latter 20 ns (top). (b) Simulated magnetization before (top) and after
(bottom) the base field is raised to $\mu_{0}H_{{\rm b}}$ (110-140 mT), and
the recipe in (a) is used. Skyrmions are annihilated in some cases (bottom
right). (c) Skyrmion annihilation rate $R_{{\rm a}}$ plotted as a function of
transverse position, $x$, across the wire for several base fields
$\mu_{0}H_{{\rm b}}$. $R_{{\rm a}}$ is determined by the number of skyrmions
in the dashed boxes in (b). The four sections in (c) divided by the cyan
dashed lines correspond to the dashed boxes in (b).

Meanwhile, we find that a slight modification of the skyrmion nucleation
recipe can be used to annihilate skyrmions, as schematically shown in Fig. 3a
along with representative MTXM images. Here, the initial state consists of
skyrmions created using the writing recipe at $\mu_{0}H$ = 94 mT (Fig. 3a:
left). The magnetic field is then changed to a specific base field
$\mu_{0}H_{{\rm b}}$ (Fig. 3a: centre). The second pulse (deleting) is then
injected to the wire, and the numbers of skyrmions before ($N_{\text{bef}}$)
and after ($N_{\text{aft}}$) the current pulse are counted. Here we emphasize
that identical current pulses are used for writing and deletion. As shown in
Fig. 3a, the number of skyrmions is reduced by $\sim 50\%$ following the
applied current pulse. Note that the skyrmion annihilation has distint spatial
asymmetry: skyrmion annihilation is more prominently on the right side of the
wire. We quantify the efficacy of deletion by the annihilation rate $R_{{\rm
a}}\left(\equiv 1-N_{\text{aft}}/N_{\text{bef}}\right)$. Fig. 3b shows a plot
of $R_{{\rm a}}$ as a function of $\mu_{0}H_{{\rm b}}$. For lower
$\mu_{0}H_{{\rm b}}$ ($<100$ mT), $R_{{\rm a}}$ maintains a nearly constant
negative value, indicating that skyrmions are instead nucleated by the current
pulse (see Fig. 1e). Above $\sim 100$ mT, $R_{{\rm a}}$ increases rapidly,
consistent with the annihilation of existing skyrmions, and reaches unity near
saturation ($\sim 130$ mT). The observation that the same current pulse drives
nucleation at lower $\mu_{0}H_{{\rm b}}$ and annihilation at higher
$\mu_{0}H_{{\rm b}}$ suggests the existence of a cross-point field
$\mu_{0}H_{{\rm c}}$ ($\sim 107$ mT), corresponding to a dynamic equilibrium
between these two phenomena.

The striking spatial asymmetry of skyrmion deletion (Fig. 3a: right), mainly
affecting the right side of the wire, is suggestive of an Oersted field driven
phenomenon. Notably, reversing the current polarity flips the spatial
asymmetry of skyrmion annihilation (SI), further supporting the Oersted field
effect. Fig. 3c shows the Oersted field profile during the current pulse,
calculated by superposing the fields generated by current segments through the
wire cross-section (SI). The Oersted field is sizable (up to $\pm 40$ mT)
compared to $\mu_{0}H_{{\rm b}}$ (80-130 mT), and introduces considerable
spatial asymmetry in the skyrmion distribution by increasing (decreasing) the
net perpendicular field on the right (left) of the wire as shown in Fig. 1e
and Fig. 3b. Near the cross-point field $\mu_{0}H_{{\rm c}}\sim 107$ mT, both
nucleation and annihilation are likely equally effective, resulting in the
clear spatial segregation of nucleation and annihilation as observed in Fig.
1d and Fig. 3a. Therefore, our scheme has distinctive merits when operating
near $\mu_{0}H_{{\rm c}}$ as one can selectively switch between nucleation and
annihilation at a given spatial location simply by reversing the current
polarity.

To elucidate the annihilation process, micromagnetic simulations were
performed following the recipe depicted in Fig. 4a (details in SI). Notably,
while a current generates an instantaneous Oersted field, generating
appreciable $\Delta T$ (c.f. Fig. 2) requires a few tens of nanoseconds 33.
Our simulation recipe accounts for this lag by using an Oersted field (profile
shown in Fig. 3c) for the entire 30 ns pulse duration, followed by the heating
phase introduced after a 10 ns delay. Fig. 4b shows simulation results at
several base fields $\mu_{0}H_{{\rm b}}$ of 110-140 mT, while Fig. 4c displays
the resulting annihilation rate $R_{{\rm a}}$ with respect to the lateral
position $x$ (defined in Fig. 4b). Consistent with experiments (Fig. 3b), the
simulated $R_{{\rm a}}$ with $\mu_{0}H_{{\rm b}}$ and the deletion is more
prominent on the right side of the wire. Meanwhile, we note that identical
current pulses used in the skyrmion writing experiments would produce similar
Oersted fields. Therefore, we have verified that such an Oersted field does
not affect the simulated nucleation results (see SI). The contrasting
influence of the Oersted field in the two cases is likely due to differences
in initial conditions and the base field magnitude.

Interestingly, the simulations allow us to analyse the annihilation process in
stepwise fashion (c.f. Fig. 4a) to disentangle the various effects at play.
The crucial finding is that all annihilation events in Fig. 4 occur within the
first 10 ns, wherein $\Delta T$ is assumed to be nearly zero (SI). No
additional annihilation is observed at elevated temperatures with or without
the Oersted field. In conjunction with the negative $\mathrm{\Delta}E$ in Fig.
2e, this suggests that Oersted field and Joule heating effects have
counteracting outcomes in (de)stabilizing skyrmions.

Figure 5: Schematic energetics for writing and deleting mechanism. (a)
Skyrmions are written onto the uniform state by injecting a current pulse at
lower $\mu_{0}H_{{\rm b}}$, wherein they are energetically favored. Joule
heating induced by the current pulse drives their nucleation by lowering the
energy barrier. (b) Deletion is effective at higher $\mu_{0}H_{{\rm b}}$ where
the uniform and skyrmion states are comparable in energy. An Oersted field
added into the base field raises the energy of the skyrmion state, and thereby
triggers annihilation of the skyrmion by injecting a current pulse.

Notably, while thermal fluctuations have also been suggested to induce
annihilation of skyrmions 34, for our case, Joule heating induces skyrmion
nucleation and prevents annihilation through rescaling of magnetic parameters
over a wide temperature range. Therefore, the time delay between the Oersted
field and heating effects, introduced to reflect experimental reality, is
crucial to enabling skyrmion deletion. Overall, while the inclusion of a more
realistic current profile and material granularity may enable better
quantitative agreement with experiments, the simulation results shown here
establish the key mechanistic insights needed to interpret the skyrmion
creation and annihilation observed in our wire devices.

## Summary and Outlook

Fig. 5a,b summarizes the energetics of skyrmion nucleation and annihilation
via Joule heating and Oersted field effects. First, at lower base fields, the
skyrmion state becomes favorable relative to the uniform state, but nucleation
is prevented by an energy barrier (Fig. 5a: left). Magnetic parameters at
elevated temperatures resulting from Joule heating reduce this barrier, and,
incidentally, further lower the skyrmion energy, thereby enabling skyrmion
nucleation (Fig. 5a: center, right). The number of skyrmions formed depends on
the energy difference of the uniform and skyrmion states, which may be
controlled by adjusting the base field. Meanwhile, at higher base fields, the
uniform state is comparable in energy with the skyrmion (Fig. 5b: left), and
the energy barrier now prevents skyrmion annihilation. However, the addition
of an Oersted field pulls up the skyrmion energy, i.e., further destabilizes
skyrmions, resulting in their annihilation regions with increased effective
field

In summary, we have described our observations of the writing and deleting of
skyrmions by lateral currents applied to a two-terminal wire device. A lateral
electric current induces two thermodynamic effects: Joule heating and magnetic
field. Our work harnesses the distinct spatial and temporal characteristics of
these effects to write and delete skyrmions with efficacies that may be tuned
by the external bias field. On one hand, Joule heating drives the nucleation
of skyrmions, inherently stable at lower fields, by modifying magnetic
parameters at elevated temperatures. On the other hand, the Oersted field
preceding Joule heating enables annihilation of skyrmions in regions of higher
fields. Together, our experiments and simulations provide a detailed energetic
picture of these mechanisms that are generalizable to other skyrmion-hosting
materials.

Our writing and deleting schemes are uniquely promising from a scalability
perspective, considering the simplicity of device design and electrical inputs
needed to implement the schemes. Their immense practical value will inspire
immediate efforts towards deterministic and field-free skyrmion manipulation
in such devices 21, 22, 23. First, while both Joule heating and Oersted field
effects may coexist and counteract, one could ensure for either of these to
dominate by varying the length and amplitude of the current pulse 21. Next,
the requisite of external fields could be realized by appropriate use of
magnetostatic effects 35, 36, 37, 38. While these schemes could be used
together in skyrmionic racetrack devices 9, 21, each scheme can also be
utilized for independently manipulating skyrmions. For example, the spatially
selective nature of Oersted field-induced annihilation could be used to
realize fabrication-free logic operations 11. Finally, Joule heating presents
itself as a promising technique to emulate a skyrmion bath or reservoir,
providing a timely experimental platform for recently proposed novel computing
architectures 12, 13.

## Methods

Micromagnetic Simulations. Simulations were performed using mumax3 software
package 29. The effective medium theory was used to examine an equivalent
reduced stack while retaining interlayer interactions 8. The magnetic
parameters were as follows (effective medium parameters in parenthesis):
$A=2.4\times 10^{-11}$ J/m ($4.0\times 10^{-12}$ J/m), $M_{{\rm S}}=1.59\times
10^{6}$ A/m ($2.64\times 10^{5}$ A/m), $K_{{\rm u}}=1.63\times 10^{6}$ J/m3
($K_{{\rm u}}=5.21\times 10^{4}$ J/m3) and $D=1.76\times 10^{-3}$ J/m2
($2.93\times 10^{-4}$ J/m2). These parameters were determined using procedures
in previous works 3, 5, 8, and are in line with published results on similar
stacks 4.

Joule Heating. Experimental determination of $\Delta T$ of the wire is
challenging due to the low thermal conductivity of the membrane (substrate)
and the short duration of the current pulse (30 ns). Therefore, $\Delta T$ is
estimated using analytical model28 with an assumption all charge is flowing
the Pt layers, and is found to be $\sim 220$ K. Correspondingly, micromagnetic
simulations were performed at elevated temperatures (up to $\sim 600$ K) to
emulate the effect of heating. The elevated temperatures were reflected in:
(a) the fluctuating thermal field in mumax3, and (b) rescaling of magnetic
parameters to reflect their values at the elevated temperatures.

Film Deposition and Device Fabrication. Multilayer stacks of Ta(4)/
[Pt(3)/Co(0.9)/MgO(1.5)]15/Ta(4)/Ru(5) (nominal layer thicknesses in nm in
parentheses) were deposited on 200 mm Si/SiO2 (20,000) wafers by ultrahigh-
vacuum magnetron sputtering at RT using a Timaris™ UHV system (base pressure
$2\times 10^{-8}$ Torr) manufactured by Singulus Technologies AG. The active
stack was repeated 15 times for optimal XMCD contrast, and the film was
simultaneously deposited on 200 nm thick X-ray transparent Si3N4 membranes for
the MTXM measurement. Nanowire devices for the MTXM measurement were
fabricated on the Si3N4 membranes using electron beam lithography (Elionix™
tool), ion-beam etching (Oxford CAIBE™), and Ta(5)/Au(100)/Ru(20) electrode
deposition (Bestec Chiron™). Completeness of resist removal was ensured using
an Axcelis™ O2 stripping tool.

Magnetization measurements were performed over 300-700 K using the EZ11
vibrating sample magnetometer (VSM) from MicroSense™ using the stacks
deposited on the Si/SiO2 substrates.

MTXM Measurements. A full-field magnetic transmission soft x-ray microscope
(MTXM; XM-1, BL6.1.2) located at the Advanced Light Source was used to image
sub-100 nm magnetic skyrmions in the nanowires. Imaging was performed in out-
of-plane geometry at the Co $L_{3}$ X-ray absorption edge ($\sim 778$ eV). A
nanosecond pulse generator (Agilent 81150A) was used to apply current pulses
to the nanowires.

Acknowledgements. Works at the ALS were supported by U.S. Department of Energy
(DE-AC02-05CH11231). S.-G.J. was supported by the National Research Foundation
of Korea (NRF, Korea) grant funded by the Korea government (MSIT)
(2020R1C1C1006194). M.-Y.I. acknowledges support by Lawrence Berkeley National
Laboratory through the Laboratory Directed Research and Development (LDRD)
Program. This work in Singapore was supported by the SpOT-LITE programme
(Grant Nos. A1818g0042, A18A6b0057), funded by Singapore’s RIE2020
initiatives, and by the Pharos Skyrmion programme (Grant No. 1527400026)
funded by A*STAR, Singapore. We also acknowledge the support of the National
Supercomputing Centre (NSCC), Singapore for computational resources. K.-S.L.
and D.-H.J. were supported by NRF of Korea grant funded MSIT
(NRF-2019R1A2C2002996, NRF-2016M3D1A1027831, and NRF-2019K1A3A7A09033400).
J.-I.H. was supported by NRF of Korea grant funded MSIT (2020R1A2C2005932).

## References

  * Nagaosa and Tokura [2013] N. Nagaosa and Y. Tokura, Topological properties and dynamics of magnetic skyrmions, Nature Nanotechnology 8, 899 (2013).
  * Soumyanarayanan  _et al._ [2016] A. Soumyanarayanan, N. Reyren, A. Fert, and C. Panagopoulos, Emergent phenomena induced by spin-orbit coupling at surfaces and interfaces, Nature 539, 509 (2016).
  * Soumyanarayanan  _et al._ [2017] A. Soumyanarayanan, M. Raju, A. L. G. Oyarce, A. K. C. Tan, M.-Y. Im, A. P. Petrović, P. Ho, K. H. Khoo, M. Tran, C. K. Gan, F. Ernult, and C. Panagopoulos, Tunable room-temperature magnetic skyrmions in Ir/Fe/Co/Pt multilayers, Nature Materials 16, 898 (2017).
  * Boulle  _et al._ [2016] O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. d. S. Chaves, A. Locatelli, T. O. Menteş, A. Sala, L. D. Buda-Prejbeanu, O. Klein, M. Belmeguenai, Y. Roussigné, A. Stashkevich, S. M. Chérif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I. M. Miron, and G. Gaudin, Room-temperature chiral magnetic skyrmions in ultrathin magnetic nanostructures, Nature Nanotechnology 11, 449 (2016).
  * Moreau-Luchaire  _et al._ [2016] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio, C. A. F. Vaz, N. V. Horne, K. Bouzehouane, K. Garcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M. George, M. Weigand, J. Raabe, V. Cros, and A. Fert, Additive interfacial chiral interaction in multilayers for stabilization of small individual skyrmions at room temperature, Nature Nanotechnology 11, 444 (2016).
  * Je  _et al._ [2020] S.-G. Je, H.-S. Han, S. K. Kim, S. A. Montoya, W. Chao, I.-S. Hong, E. E. Fullerton, K.-S. Lee, K.-J. Lee, M.-Y. Im, and J.-I. Hong, Direct Demonstration of Topological Stability of Magnetic Skyrmions via Topology Manipulation, ACS Nano 14, 3251 (2020).
  * Jiang  _et al._ [2015] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B. Jungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. t. Velthuis, and A. Hoffmann, Blowing magnetic skyrmion bubbles, Science 349, 283 (2015).
  * Woo  _et al._ [2016] S. Woo, K. Litzius, B. Krüger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer, M. Kläui, and G. S. D. Beach, Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic ferromagnets, Nature Materials 15, 501 (2016).
  * Fert  _et al._ [2017] A. Fert, N. Reyren, and V. Cros, Magnetic skyrmions: advances in physics and potential applications, Nature Reviews Materials 2, 17031 (2017).
  * Parkin  _et al._ [2008] S. S. P. Parkin, M. Hayashi, and L. Thomas, Magnetic Domain-Wall Racetrack Memory, Science 320, 190 (2008).
  * Kang  _et al._ [2016] W. Kang, Y. Huang, X. Zhang, Y. Zhou, and W. Zhao, Skyrmion-Electronics: An Overview and Outlook, Proceedings of the IEEE 104, 2040 (2016).
  * Pinna  _et al._ [2018] D. Pinna, F. A. Araujo, J.-V. Kim, V. Cros, D. Querlioz, P. Bessiere, J. Droulez, and J. Grollier, Skyrmion Gas Manipulation for Probabilistic Computing, Physical Review Applied 9, 064018 (2018).
  * Prychynenko  _et al._ [2018] D. Prychynenko, M. Sitte, K. Litzius, B. Krüger, G. Bourianoff, M. Kläui, J. Sinova, and K. Everschor-Sitte, Magnetic Skyrmion as a Nonlinear Resistive Element: A Potential Building Block for Reservoir Computing, Physical Review Applied 9, 014034 (2018).
  * Iwasaki  _et al._ [2013a] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Current-induced skyrmion dynamics in constricted geometries, Nature Nanotechnology 8, 742 (2013a).
  * Lin  _et al._ [2013a] S.-Z. Lin, C. Reichhardt, and A. Saxena, Manipulation of skyrmions in nanodisks with a current pulse and skyrmion rectifier, Applied Physics Letters 102, 222405 (2013a).
  * Lin  _et al._ [2013b] S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena, Driven skyrmions and dynamical transitions in chiral magnets, Physical Review Letters 110, 207202 (2013b).
  * Sampaio  _et al._ [2013] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nucleation, stability and current-induced motion of isolated magnetic skyrmions in nanostructures, Nature Nanotechnology 8, 839 (2013).
  * Miron  _et al._ [2011] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection, Nature 476, 189 (2011).
  * Iwasaki  _et al._ [2013b] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Universal current-velocity relation of skyrmion motion in chiral magnets, Nature Communications 4, 1463 (2013b).
  * Hrabec  _et al._ [2017] A. Hrabec, J. Sampaio, M. Belmeguenai, I. Gross, R. Weil, S. M. Chérif, A. Stashkevich, V. Jacques, A. Thiaville, and S. Rohart, Current-induced skyrmion generation and dynamics in symmetric bilayers, Nature Communications 8, 15765 (2017).
  * Finizio  _et al._ [2019] S. Finizio, K. Zeissler, S. Wintz, S. Mayr, T. Weßels, A. J. Huxtable, G. Burnell, C. H. Marrows, and J. Raabe, Deterministic Field-Free Skyrmion Nucleation at a Nanoengineered Injector Device, Nano Letters 19, 7246 (2019).
  * Büttner  _et al._ [2017] F. Büttner, I. Lemesh, M. Schneider, B. Pfau, C. M. Günther, P. Hessing, J. Geilhufe, L. Caretta, D. Engel, B. Krüger, J. Viefhaus, S. Eisebitt, and G. S. D. Beach, Field-free deterministic ultrafast creation of magnetic skyrmions by spin–orbit torques, Nature Nanotechnology 12, 1040 (2017).
  * Woo  _et al._ [2018] S. Woo, K. M. Song, X. Zhang, M. Ezawa, Y. Zhou, X. Liu, M. Weigand, S. Finizio, J. Raabe, M.-C. Park, K.-Y. Lee, J. W. Choi, B.-C. Min, H. C. Koo, and J. Chang, Deterministic creation and deletion of a single magnetic skyrmion observed by direct time-resolved X-ray microscopy, Nature Electronics 1, 288 (2018).
  * Juge  _et al._ [2019] R. Juge, S.-G. Je, D. d. S. Chaves, L. D. Buda-Prejbeanu, J. Peña-Garcia, J. Nath, I. M. Miron, K. G. Rana, L. Aballe, M. Foerster, F. Genuzio, T. O. Mentes, A. Locatelli, F. Maccherozzi, S. S. Dhesi, M. Belmeguenai, Y. Roussigné, S. Auffret, S. Pizzini, G. Gaudin, J. Vogel, and O. Boulle, Current-Driven Skyrmion Dynamics and Drive-Dependent Skyrmion Hall Effect in an Ultrathin Film, Physical Review Applied 12, 044007 (2019).
  * Chao  _et al._ [2005] W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, Soft X-ray microscopy at a spatial resolution better than 15 nm, Nature 435, 1210 (2005).
  * Woo  _et al._ [2017] S. Woo, K. M. Song, H.-S. Han, M.-S. Jung, M.-Y. Im, K.-S. Lee, K. S. Song, P. Fischer, J.-I. Hong, J. W. Choi, B.-C. Min, H. C. Koo, and J. Chang, Spin-orbit torque-driven skyrmion dynamics revealed by time-resolved X-ray microscopy, Nature Communications 8, 15573 (2017).
  * Lemesh  _et al._ [2018] I. Lemesh, K. Litzius, M. Böttcher, P. Bassirian, N. Kerber, D. Heinze, J. Zázvorka, F. Büttner, L. Caretta, M. Mann, M. Weigand, S. Finizio, J. Raabe, M. Im, H. Stoll, G. Schütz, B. Dupé, M. Kläui, and G. S. D. Beach, Current‐Induced Skyrmion Generation through Morphological Thermal Transitions in Chiral Ferromagnetic Heterostructures, Advanced Materials 30, 1805461 (2018).
  * Fangohr  _et al._ [2011] H. Fangohr, D. S. Chernyshenko, M. Franchin, T. Fischbacher, and G. Meier, Joule heating in nanowires, Physical Review B 84, 054437 (2011).
  * Vansteenkiste  _et al._ [2014] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. V. Waeyenberge, The design and verification of MuMax3, AIP Advances 4, 107133 (2014).
  * Nembach  _et al._ [2015] H. T. Nembach, J. M. Shaw, M. Weiler, E. Jué, and T. J. Silva, Linear relation between Heisenberg exchange and interfacial Dzyaloshinskii–Moriya interaction in metal films, Nature Physics 11, 825 (2015).
  * Moreno  _et al._ [2016] R. Moreno, R. F. L. Evans, S. Khmelevskyi, M. C. Muñoz, R. W. Chantrell, and O. Chubykalo-Fesenko, Temperature-dependent exchange stiffness and domain wall width in Co, Physical Review B 94, 104433 (2016).
  * Tomasello  _et al._ [2018] R. Tomasello, K. Y. Guslienko, M. Ricci, A. Giordano, J. Barker, M. Carpentieri, O. Chubykalo-Fesenko, and G. Finocchio, Origin of temperature and field dependence of magnetic skyrmion size in ultrathin nanodots, Physical Review B 97, 060402 (2018).
  * Kim  _et al._ [2008] K.-J. Kim, J.-C. Lee, S.-B. Choe, and K.-H. Shin, Joule heating in ferromagnetic nanowires: Prediction and observation, Applied Physics Letters 92, 192509 (2008).
  * Legrand  _et al._ [2017] W. Legrand, D. Maccariello, N. Reyren, K. Garcia, C. Moutafis, C. Moreau-Luchaire, S. Collin, K. Bouzehouane, V. Cros, and A. Fert, Room-Temperature Current-Induced Generation and Motion of sub-100 nm Skyrmions, Nano Letters 17, 2703 (2017).
  * Zeissler  _et al._ [2017] K. Zeissler, M. Mruczkiewicz, S. Finizio, J. Raabe, P. M. Shepley, A. V. Sadovnikov, S. A. Nikitov, K. Fallon, S. McFadzean, S. McVitie, T. A. Moore, G. Burnell, and C. H. Marrows, Pinning and hysteresis in the field dependent diameter evolution of skyrmions in Pt/Co/Ir superlattice stacks, Scientific Reports 7, 15125 (2017).
  * Juge  _et al._ [2018] R. Juge, S.-G. Je, D. d. S. Chaves, S. Pizzini, L. D. Buda-Prejbeanu, L. Aballe, M. Foerster, A. Locatelli, T. O. Menteş, A. Sala, F. Maccherozzi, S. S. Dhesi, S. Auffret, E. Gautier, G. Gaudin, J. Vogel, and O. Boulle, Magnetic skyrmions in confined geometries: Effect of the magnetic field and the disorder, Journal of Magnetism and Magnetic Materials 455, 3 (2018).
  * Ho  _et al._ [2019] P. Ho, A. K. Tan, S. Goolaup, A. G. Oyarce, M. Raju, L. Huang, A. Soumyanarayanan, and C. Panagopoulos, Geometrically Tailored Skyrmions at Zero Magnetic Field in Multilayered Nanostructures, Physical Review Applied 11, 024064 (2019).
  * Finocchio  _et al._ [2016] G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, and M. Kläui, Magnetic skyrmions: from fundamental to applications, Journal of Physics D: Applied Physics 49, 423001 (2016).

Supplementary Information

## S1. Simulating Joule Heating Effects

Figure S1: Simulations of Joule heating-driven skyrmion writing. (a) Schematic
of the simulation recipe used at a base field of 90 mT to emulate a
temperature rise $\Delta T$ for the current pulse duration, with rescaling of
magnetic parameters (see main text) during the pulse. Labels A to E identify
various instances within the simulation recipe. (b-e) Grayscale snapshots of
simulated out-of-plane magnetization at instances B to E [defined in (a)] for
(b) $\Delta T$ = 111 K, (c) $\Delta T$ = 165 K, (d) $\Delta T$ = 165 K
(without rescaling of magnetic parameters) and (e) $\Delta T$ = 228 K. While B
is identical for all cases, the intermediate (C, D) and final (E)
configurations vary, and show skyrmion formation in some cases.

Joule heating recipe. Micromagnetic simulations of skyrmion writing follow the
Joule heating simulation recipe shown in Fig. S1a. The system was initialized
to at positive saturation (Step A) and lowered to the base field (90 mT for
Fig. S1) at 300 K while retaining the uniformly magnetized state, and allowed
to equilibrate for 5 ns (Step B). The temperature was then elevated by $\Delta
T$ for 30 ns (Step C), corresponding to the pulse duration. During this time,
the micromagnetic parameters, $M_{{\rm S}}$, $D$, _$K_{{\rm u}}$_ , and _$A$_
were rescaled to values corresponding to $T+\Delta T$ following scaling
relations detailed in the main text. Subsequently, the temperature was reset
to 300 K, and the magnetic parameters were restored to their RT values (Step
D). The system was then allowed to equilibrate at RT for 15 ns to produce the
final state (Step E).

Nucleation threshold. First, while the final state for Fig. S1b ($\Delta T$ =
111 K) remains uniformly magnetized, Fig. S1c ($\Delta T$ = 165 K) shows
nucleated skyrmions. This supports the existence of a threshold value of
$\Delta T$ (between 111-165 K) for skyrmion nucleation via Joule heating.
Second, as seen in Fig. S1d, no skyrmions are nucleated when the $\Delta T$ =
165 K simulations are performed without rescaling magnetic parameters. This
suggests that the rescaling of magnetic parameters, which likely lowers the
energy barrier, is critical to this skyrmion nucleation mechanism.

Nucleation snapshots. Fig. S1c shows that skyrmions are nucleated at elevated
temperatures for $\Delta T$ = 165 K (Step C). When $\Delta T$ is removed, the
configuration remains largely the same (Step D), and further equilibration
affects a small fraction of the textures (Step E). However, this is not the
case in Fig. S1e, wherein $\Delta T$ is increased to 228 K. In this case,
large fluctuations at elevated temperatures produce a randomly magnetized
configuration (Step C). The removal of $\Delta T$ may arbitrarily create a few
nucleation (Step D), which equilibrate to form skyrmions (Step E). While both
Fig. S1c and Fig. S1e produce skyrmion configurations, their distinction is
crucial. In case of Fig. S1e parameters, successive current pulses should
correspond to drastically different magnetic configurations. In contrast, they
may produce only incremental changes for Fig. S1b parameters.

## S2. Simulating Oersted Field Effects

Figure S2: Simulations of Oersted field-driven skyrmion deletion. (a)
Schematic of simulation recipe used to simulate Oersted field effects at base
field $\mu_{0}H_{2}$ with a temperature rise $\Delta T=$165 K (as in Fig. S1.
Labels F to J identify various instances within the recipe. The Oersted field
is applied for the entire 30 ns pulse duration (starting at F), while Joule
heating is introduced for the final 20 ns (starting at G). (b-c) Grayscale
snapshots of simulated magnetization at instances F to J [defined in (a)] for
(b) $\mu_{0}H_{2}=$ 120 mT and (c) $\mu_{0}H_{2}=$ 130 mT, showing fewer
skyrmions in J (c.f. F).

Oersted Field Simulation Recipe. Micromagnetic simulations of skyrmion
deletion follow the Oersted field effect simulation recipe shown in Fig. S2a.
The system is initialized with the skyrmion configuration created in Fig. S1a
Step E. The field is raised to $\mu_{0}H_{2}$, and allowed to equilibrate for
15 ns (Step F). At this point, the application of a current pulse results in
an instantaneous Oersted field and a delayed Joule heating effect. These are
simulated by applying an Oersted field with profile in Figure 3c for the full
current pulse duration and elevating the temperature by $\Delta T$ [as in Fig.
S1a] after the pulse has been applied for 10 ns. This corresponds first to a
10 ns time window with an Oersted field at 300 K (Step G), and then to a 20 ns
time window where the Oersted field and elevated temperature are
simultaneously in effect (Step H). Subsequently both these effects are turned
off as the system is restored to RT (Step I), followed by equilibration for 15
ns to give the final state (Step J).

Annihilation Snapshots. Skyrmion annihilation is seen for both Fig. S2b
($\mu_{0}H_{2}=120$ mT) and Fig. S2c ($\mu_{0}H_{2}=130$ mT): the final state
(J) has fewer skyrmions than the initial state (F). As clarified in the main
text, the annihilation is spatially asymmetric: skyrmions are mostly
annihilated on the right side of the image. Here, we clarify the annihilation
window by examining the magnetization snapshots across the simulated duration.
From inspection of Fig. S2b and c, it is evident that the annihilation occurs
predominantly in the 10 ns (between F and G) wherein the Oersted field is
present at RT. In contrast, the comparison of snapshots after G, wherein the
temperature is also elevated, reveal negligible annihilation, albeit with some
random motion of skyrmions. Above statement can be more clearly seen in Fig.
S2d. we have tracked the number of skyrmions at each step of the skyrmion
deletion simulation at different fields. Fig. S2d shows the number of deleted
skyrmions during the Oersted field only phase (F-G) and the heating phase
(G-H). It is evident that, during the heating phase (G-H), there is no
decrease in the number of skyrmions, and the decrease in the number of
skyrmions only takes place during the Oersted field phase (F-G). This suggests
that Joule heating effects counteract Oersted field deletion by stabilizing
skyrmions.

Figure S3: Simulations of Oersted field effects on skyrmion nucleation. (a)
Simulation recipes for Joule heating-driven skyrmion nucleation starting from
a uniform state without Oersted fields. (b) Simulation recipes for Joule
heating-driven skyrmion nucleation with Oersted fields. Labels A to E identify
various instances within the two recipes. (c-d) Grayscale snapshots of
simulated magnetization at instances A to E [defined in (a)] for (c) no
Oersted field, and (d) with Oersted field.

We note that Joule heating may have been associated with skyrmion erasure in
some previous works. In our case, through systematic simulations detailed in
Fig. S1, we have found that, over a reasonably wide temperature ranges, the
rescaling of micromagnetic parameters associated with heating stabilizes
existing skyrmions, and in fact prefers to nucleate additional skyrmions. This
allowed us to conclude that the Oersted field is the main factor for the
skyrmion annihilation in our case. In this light, we conclude that works
reporting heat-based erasure of skyrmions may have explored a rather different
phase space regime.

Oersted fields & nucleation. As the nucleation and annihilation pulses are
identical, their effects, i.e. Joule heating and Oersted field, should appear
in both cases. Having considered them in conjunction during annihilation, we
now revert to verifying if the introduction of Oersted fields affects
nucleation. In particular, we compare the results from the Fig. S1a recipe (no
Oersted field) with those from the Fig. S2a recipe (with Oersted field) for a
uniformly magnetized initial state with $\Delta T$ = 165 K (see Fig. S3).
Notably we find little difference between the two sets of results shown in
Fig. S3b-c in the intermediate or final stages. Both produce skyrmion
configurations with similar characteristics. The diminished contribution of
Oersted field effect for this case is due to the uniformly magnetized state in
the absence of Joule heating effects (before B). This suggests that once
skyrmions are created at elevated temperatures (before C), their deletion is
prevented by the concomitant role of Joule heating. In summary, this validates
the simulation protocol followed in Fig. S1 (and Fig. 2 in the main text).

## S3. Pinning Effects on Skyrmion Nucleation

Figure S4: Spatial distribution of the observed skyrmion nucleation obtained
by overlaying 11 MTXM images recorded after applying current pulses to the
uniformly magnetized wire device. Red dashed lines indicate the wire edges.
The contrast outside of the wire originates from irregularities in the
transparent membrane.

Within multilayer films, we expect that pinning effects may arise from grain
boundaries. These should lead to higher resistivity and therefore to greater
heating effects for the same current density. However, from the
Mayadas–Shatzkes model [Mayadas et al. Phys. Rev. B 1. 1382 (1970)], electron
scattering from grain boundaries is only one of the possible resistive
mechanisms. In ultrathin layered films ($\sim 1$ nm per layer), the dominant
mechanism might be interface scattering [Misra et al. J. Appl. Phys. 85, 302
(1999)]. If this is the case, the correlation between pinning and heating
effects will be weak. In order to experimentally address this question, one
may need to controllably vary the grain size, e.g., by tuning deposition
pressure or performing post-annealing, and subsequently measure the residual
resistivity ratio (RRR) of the resulting films. Such experiments are beyond
the scope of this manuscript, but nevertheless are important directions for
follow-up work.

Figure S5: Micromagnetic simulation for the presence of grains. Simulation
recipe for the skyrmion nucleation (a) and overlaid results from 8 simulations
without grains (b) and with grains (c), with grain boundaries shown in (d).

Notwithstanding their contribution to heating, the question of whether pinning
effects are indeed occurring is equally relevant. To address this, we have
investigated the spatial distribution of skyrmion nucleation probability in
our device for both experimental and simulations. Fig. S4 shows the result
obtained by overlapping 11 skyrmion nucleation images. There are several
darker spots, corresponding to a high skyrmion nucleation probability. This
indicates that our nucleation scheme depends on the local variation of
material characteristics, such as defects, which could also serve as pining
sites. However, at the moment, we cannot confirm whether the high nucleation
probability sites correspond to pinning sites, as this would require
investigating skyrmion motion and the distribution of the pinning sites.

Meanwhile, to check if pinning effect could play a role in skyrmion
nucleation, we have performed additional simulations with the introduction of
random grains. We have compared the results from two sets of simulations
(repeated 8 times) with and without grains respectively. For the latter case,
the uniaxial anisotropy and interfacial DMI are varied by $10\%$ while all
other magnetic parameters remain the same across grains with average size
$\sim 200$ nm. Fig. S5 shows the overlaid distributions of nucleated skyrmions
obtained at the final relaxed stage E in Fig. S5a without grains (b) and with
grains (c). It is apparent that, with the introduction of grains, skyrmion
nucleation occurs more frequently in specific grains, and the skyrmions are
pinned at the edges of the grains (d). This result probably explains the
several darkest spots in Fig. S4, reflecting the presence of grains strongly
affects the nucleation positions of skyrmions.

## S4. Skyrmion Size Analysis

Figure S6: Analysis on skyrmion size with respect to the perpendicular field.
(a) Analysis on the images from Figure 1d. (b) Analysis on the results from
the quasi-static experiments.

We performed analysis on the images in manuscript Fig. 1d for the nucleated
skyrmion size (Fig. S6a). Different from the decreasing trend of the skyrmion
size with increasing field, the data does not show a decrease in the skyrmion
size with the magnetic field.

We to note that the X-ray images in manuscript Fig. 1d are not optimized to
precisely determine skyrmion sizes, even though the data quality is sufficient
to determine the presence of skyrmions. We further caution that the
interpretation of size trends is constrained by limited nucleation statistics
at higher fields (10 times lower, see Fig. S6a top). In summary, the nucleated
skyrmion images does not have sufficient statistical significance to make
meaningful claims about skyrmion size trends with magnetic field.

However, to clarify the evolution of skyrmion size with magnetic field, we
performed additional experiments with optimized microscopy conditions. The
results in Fig. S6b show a pronounced decrease in skyrmion size with
increasing magnetic fields in line with the simulation result in manuscript
Fig. 2c.

Please note that, in this additional experiment, initial skyrmions were
nucleated at the low field, and we tracked their size as increasing the field.
This quasi-static experimental protocol is different from the experiments
shown in manuscript Fig. 1d where skyrmions are distinctly nucleated at each
measured field. Therefore, we cannot completely exclude the possibility that
the sizes of the skyrmions from the two cases cannot be matched. Notably, one
could have an inconsistency in skyrmion sizes if only bigger skyrmions survive
the current pulse induced thermal fluctuations once nucleated at higher field.
However, in the quasi-static case, small skyrmions may also survive in higher
fields because there are no significant thermal fluctuations. The effect of
pinning could also influence the two cases in different ways.

## S5. Calculation of the Oersted Field

Figure S7: (a) Perpendicular component of the Oersted field profile calculated
for various sizes of the wire segments and (b) comparison between calculated
Oersted field profile (perpendicular) and that from the Ampère’s circuital law
for a straight line.

The Oersted field in the manuscript was calculated by superposing the magnetic
fields generated from wire segments (1 nm2 cross-section) through the wire
cross-section (2 $\mu$m $\times$ 58 nm). Uniform current density was assumed
and the Biot-Savart Law was used. The wire segments of 1 nm cell were chosen
because the cell-size dependence of calculated magnetic field profile
disappears as the cell-size gets below 10 nm as shown in Fig. S7a. To test the
validity of our code, the calculated magnetic field profile is compared with
the result of the well-known equation $B=(\mu_{0}/2\pi)\cdot I\,/\,x$, the
magnetic field from a straight line (Ampère’s circuital law), as shown in Fig.
S7b. It shows perfect agreement with the equation outside of the wire ($|x|>1$
$\mu$m), confirming that our code for the Oersted field calculation is
correct.

## S6. Skyrmion Annihilation for Reversed Field and Current

Figure S8: Skyrmion annihilation by the application of a (a) positive current
pulse and (b) negative current pulse (scale bar: 1 $\mu$m).

To check if the spatial asymmetry of the skyrmion distribution in the
annihilation experiment indeed originates from the Oersted field, we performed
skyrmion annihilation experiments with reversed current pulse. The result is
displayed in Fig. S8b and is obtained from a different wire but on the same
film. The images in Fig. S8a are taken from manuscript Fig. 3a for comparison.
It is evident that the skyrmion annihilation is dominant on the left side of
the wire for the negative pules (Fig. S8b) while the annihilation is dominant
on the right side of the wire for the positive current pules (Fig. S8a). These
results are in perfect agreement with the presence Oersted field, confirming
the Oersted field effect.