††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Robust scenario for the generation of non-equilibrium topological
fluctuation states

Kathinka Gerlinger  Max Born Institute for Nonlinear Optics and Short Pulse
Spectroscopy, 12489 Berlin, Germany  Rein Liefferink  Radboud University,
Institute for Molecules and Materials, 6525 AJ Nijmegen, the Netherlands
Michael Schneider  Max Born Institute for Nonlinear Optics and Short Pulse
Spectroscopy, 12489 Berlin, Germany  Lisa-Marie Kern  Max Born Institute for
Nonlinear Optics and Short Pulse Spectroscopy, 12489 Berlin, Germany
Christopher Klose  Max Born Institute for Nonlinear Optics and Short Pulse
Spectroscopy, 12489 Berlin, Germany  Daniel Metternich  Helmholtz-Zentrum
Berlin, 14109 Berlin, Germany  Dieter Engel  Max Born Institute for Nonlinear
Optics and Short Pulse Spectroscopy, 12489 Berlin, Germany  Flavio Capotondi
Elettra Sincrotrone Trieste, 34149 Basovizza TS, Italy  Dario De Angelis
Elettra Sincrotrone Trieste, 34149 Basovizza TS, Italy  Matteo Pancaldi
Elettra Sincrotrone Trieste, 34149 Basovizza TS, Italy  Emanuele Pedersoli
Elettra Sincrotrone Trieste, 34149 Basovizza TS, Italy  Felix Büttner
Helmholtz-Zentrum Berlin, 14109 Berlin, Germany  Stefan Eisebitt  Max Born
Institute for Nonlinear Optics and Short Pulse Spectroscopy, 12489 Berlin,
Germany  Technische Universität Berlin, Institut für Optik und Atomare Physik,
10623 Berlin, Germany  Johan H. Mentink<EMAIL_ADDRESS>Radboud
University, Institute for Molecules and Materials, 6525 AJ Nijmegen, the
Netherlands  Bastian Pfau<EMAIL_ADDRESS>Max Born Institute for
Nonlinear Optics and Short Pulse Spectroscopy, 12489 Berlin, Germany

###### Abstract

The recently discovered topological fluctuation state provides a fascinating
new perspective on the ultrafast emergence of topology in condensed matter
systems. However, rather little is known about the physics of this state and
the origin of the topological fluctuations. Using time-resolved small-angle
x-ray scattering, we observe that topological fluctuation states appear after
laser excitation even if the final state does not host stable skyrmions.
Simulations support these findings and reveal that the fluctuations originate
from the competition between spontaneous nucleation and decay of skyrmions,
consistent with Arrhenius-like activation over a potential barrier. Stable
skyrmions can freeze out of such fluctuations when the effective temperature
of the system relaxes faster than the decay time of the skyrmions. Our results
reveal a robust scenario for the generation of topological fluctuation states,
potentially enabling their study in a wide variety of magnetic systems.

The emergence of nanometer-scale magnetic textures in thin-film magnetic
multilayers with perpendicular anisotropy has fascinated scientists for a long
time and stimulated widespread fundamental and applied research on these
customizable materials [1, 2]. This research has even intensified in the last
years by the discovery of topological textures, so-called skyrmions, which
stabilize in the material, even at room temperature, by competing interactions
[3, 4, 5, 6]. Typically, however, the skyrmion state remains hidden during
adiabatic field cycling [5] due to the large topological energy barrier
associated with its nucleation.

The most common way to create skyrmions is the application of spin torques
generated from spin-polarized currents [7, 5, 8, 9]. Recently, though, it was
demonstrated that also ultrashort laser excitation leads to the formation of a
skyrmion state in the material [10, 11] if the laser fluence exceeds a certain
nucleation threshold [11, 12]. This topological phase transition proceeds on a
timescale of only 300 ps, despite the massive reorientation of spins required
to nucleated areas of switched magnetization in the ferromagnet [11]. Even
more important, this research provided the first evidence of the existence of
a so-called fluctuation state which mediates the topological switching.
Previous equilibrium Monte-Carlo simulations [13, 14], have elucidated similar
fluctuating states in the equilibrium phase diagram, localized between field-
polarized ferromagnetic and paramagnetic phases. Still, very little is known
about the physics of topological fluctuations states out of equilibrium,
particularly their role in topological phase transitions and the fundamental
speed limit of such transitions. For example, previous work explained the
topological switching in the fluctuation state mainly by an effective
elimination of the topological energy barrier [11]. However, it remains
unclear how the symmetry breaking towards stable skyrmions proceeds
dynamically and if this process necessitates non-equilibrium conditions.

Here, we report on details of the fluctuation state accessible in Co/Pt
multilayers after laser excitation. Combining resonant scattering experiments
and atomistic spin simulations on ultrafast timescales, we find that the
system robustly passes through the topological fluctuation state—even when the
topology of the final state is trivial. We identify the fluctuation state as a
competition between nucleation and decay of skyrmions. Its dynamics can be
explained with Arrhenius-like activation over an energy barrier over a broad
range of temperatures. Based on field and temperature-dependent relaxation
times, we model the topological fluctuations with a basic rate equation, even
under highly non-equilibrium conditions and on timescales much shorter than
typical micromagnetic timescales. As a result, stable skyrmions form when the
effective temperature of the system relaxes faster than the decay time of the
skyrmions, opening up new possibilities to approach the speed limit for the
topological phase transition.

Figure 1: Detection of the topological magnetic states. (a) Illustration of
the pump–probe setup. (b) The out-of-plane magnetic hysteresis of the
multilayer film. The blue circles indicate the fields at which time-resolved
data were acquired. (c,d) X-ray microscopy images of the skyrmion state after
exposure with one IR pulse at an applied field of $30\text{\,}\mathrm{mT}$ and
$60\text{\,}\mathrm{mT}$, respectively. (e,f) Scattered intensity as function
of momentum transfer $q$ at delay times indicated and under an applied field
of $34\text{\,}\mathrm{mT}$ and $197\text{\,}\mathrm{mT}$, respectively. For
better comparability, the spectra at $400\text{\,}\mathrm{ps}$ and
$2.2\text{\,}\mathrm{ns}$ in (e) were divided by 4 and 10, respectively.

Our magnetic multilayer sample consists of $12$ repeats of
Co($0.6\text{\,}\mathrm{n}\mathrm{m}$)/Pt($0.8\text{\,}\mathrm{n}\mathrm{m}$)
bilayers and exhibits a typical sheared hysteresis of a thin film ferromagnet
with perpendicular anisotropy, forming lateral magnetic domains stabilized by
stray fields (Fig. 1(b), see supplemental material (SM), section I). In such
multilayers, excitation with a single ultrashort laser pulse creates a
skyrmion state [10, 11] with a density controlled by the applied magnetic
field [12]. We confirmed this for our sample via static x-ray microscopy
measurements (Fig. 1(c,d), SM section I, and Ref. 15).

To detect the transient fluctuation state, we performed time-resolved
measurements by probing lateral spin textures with resonant small-angle x-ray
scattering (SAXS) in transmission after IR excitation at
$795\text{\,}\mathrm{nm}$ wavelength with $65\text{\,}\mathrm{fs}$ (fwhm)
pulses (Fig. 1(a)). Importantly, we varied the applied fields during the
measurements (indicated as blue dots in Fig. 1(b)), corresponding to different
skyrmion densities in the final states down to saturated states. Experiments
were conducted at the DiProI beamline of the free-electron laser (FEL) source
FERMI@Elettra, providing sub-$100\text{\,}\mathrm{fs}$ (fwhm) pulses of XUV
radiation [16]. To establish resonant scattering conditions, the FEL was tuned
to the Co M2,3 edge ($20.8\text{\,}\mathrm{n}\mathrm{m}$ wavelength),
resulting in a sensitivity to out-of-plane magnetic moments (see SM section I
for details). As a typical experimental procedure, we saturated the sample at
$\approx 220\text{\,}\mathrm{mT}$ in between two FEL shots and set the desired
external field for the subsequent pump–probe measurement in an automated field
cycling synchronized to the FEL repetition rate of $50\text{\,}\mathrm{Hz}$
(see supplemental Fig. S1).

Azimuthal integrations of typical scattering patterns recorded at different
delay times are shown in Fig. 1(e,f) for $34\text{\,}\mathrm{mT}$ and
$197\text{\,}\mathrm{mT}$, respectively, as a function of the momentum
transfer $q$ (see supplemental Fig. S4 for all integrations). The magnetic
film and the 30-nm thick silicon-nitride substrate are laterally almost
homogeneous and scattering only originates from lateral variations of the
magnetization.

For both fields shown in Fig. 1(e,f), we observe that
$100\text{\,}\mathrm{ps}$ after IR excitation, the scattering spans across a
large range in momentum space, in particular indicating the presence of short-
range spin correlations during the fluctuation state [11]. At these early
delay times, we observe that the shape of the SAXS spectra is almost
independent of the applied magnetic field, with similar total scattering
intensity. At later times, however, the evolution of the scattering patterns
is very different depending on the applied field. At low field, the scattering
intensity distribution narrows and shifts its maximum towards lower scattering
vectors. At the same time, the intensity rapidly increases, indicating the
progressing formation and growth of skyrmions. The very prominent scattering-
intensity maximum of the resulting final state at delays
$>1\text{\,}\mathrm{ns}$ reveals the creation of a skyrmion pattern with a
high degree of long-range ordering in agreement with the corresponding real-
space image (Fig. 1(c)). In contrast, for a high applied field, the sample
ultimately returns to the uniform saturated state, which is evidenced by a
complete loss of scattering intensity.

Figure 2: Results from the pump–probe scattering experiments. (a) The total
scattered intensity $I$ and (b) the correlation length
($\lambda_{C}=2\pi/q_{0}$), for six different applied magnetic fields as a
function of the delay between IR pump and FEL probe.

The entire picosecond evolution of the total scattering intensity is shown in
Fig. 2(a) as a function of the pump–probe delay for different applied fields.
Notably, in the first tens of picoseconds after excitation, the scattered
intensity increases with a rate that is almost independent of the applied
field. At lower fields, promoting a stable final skyrmion state, the intensity
keeps growing also on time scales of several hundreds of picoseconds. On the
other hand, the scattering intensity again completely declines after a maximum
at around $150\text{\,}\mathrm{ps}$ for higher fields where the film finally
transitions back to the saturated state. The large variation of scattering
intensity in the final states directly reflects the field dependence of the
skyrmion density [12].

In the next step, we deduce an estimate for the dominant length scales of the
lateral spin correlations causing the emerging magnetic scattering intensity.
A common measure for this length scale is the correlation length—the inverse
$q$-space position of the SAXS intensity maximum ($q_{0}$), plotted in Fig.
2(b). Once an intensity maximum has formed after $50\text{\,}\mathrm{ps}$, the
correlation length steadily increases with time. It is remarkable that this
evolution is almost entirely independent of the applied field, in particular
at early times ($<300\text{\,}\mathrm{ps}$).

Despite the high similarity of all SAXS spectra at early times
($<300\text{\,}\mathrm{ps}$), the SAXS spectra at late times exhibit
significant differences at higher momentum transfer—a region where the form
factor of the skyrmions dominates the scattering (Fig. 1(e,f)). We fitted the
average form factor assuming circular textures with normally distributed sizes
to the SAXS spectra of the three final states containing skyrmions (SM,
section II, Fig. S6). The radius decreases with the applied magnetic field as
is expected from theory [17] and previous measurements on similar samples [12,
18].

It is already known that the early times of the skyrmion formation process are
governed by spin and topological fluctuations while later times are
characterized by a relaxation of the state to the final skyrmion size and
density [11]. We, here, again clearly detect this fluctuation state based on
the evolution of the scattering intensity and in-plane correlation length. The
behavior at early times ($<300\text{\,}\mathrm{ps}$) shows that the magnetic
film transiently enters the topological fluctuation state no matter if a
stable skyrmion state is finally formed or not. The final formation of
skyrmions is, thus, not a necessary condition for the transient occurrence of
the fluctuation state. Under suitable conditions, transient topological
fluctuations do exist even if ultimately the topology remains unchanged. Only
at later times ($>300\text{\,}\mathrm{ps}$), the applied field governs the
relaxation process and the resulting final state.

Figure 3: Results from the atomistic spin simulations. (a)
$100100\text{\,}\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{s}$ snapshots
of $M_{iz}$ at
$10\text{\,}\mathrm{ps}30\text{\,}\mathrm{ps}100\text{\,}\mathrm{ps}$ as
indicated in (c) and (d) at $0.5\text{\,}\mathrm{T}$ (i–iii, circles) and
$1.25\text{\,}\mathrm{T}$ (iv–vi, triangles). (b) Time evolution of the
topological charge $Q$ in the images shown in (a). Time evolution of (c) the
intensity $I$ of the Fourier transform of $M_{iz}$ and (d) the topological
charge $Q$, mean from $10$ simulation runs of
$500500\text{\,}\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{s}$ at
different magnetic fields. The temperature pulse is shown as a dotted gray
line in (c). The dashed lines in (d) show the predictions of the topological
charge obtained from a rate-equation model.

We will interpret these results using atomistic simulations, assuming a
minimal model for supporting skyrmions with only short-range interactions,
which was previously found to be sufficient to qualitatively understand the
short-time nucleation dynamics [11] (more details in SM section III). We
performed simulations at varying magnetic fields with a temperature pulse to
mimic the laser excitation (shown as a gray dotted line in Fig. 3(c)). Example
snapshots from the simulation of the spin’s z-component ($M_{iz}$) with
progressing simulation time are shown in Fig. 3(a) for the lowest and highest
magnetic field used. In agreement with the experiment, the system develops
skyrmions at the lowest field, while at high applied field, the system returns
to a saturated state.

The azimuthally integrated spectra of the intensity of the Fourier transform
$\mathcal{F}$ of $M_{iz}$ (shown in the supplemental Fig. S7) behave similarly
to the experimental data (Fig. 1(f,g)) but are not identical in shape due to
the lack of long-range interactions. Fig. 3(c) shows the sum of
$|\mathcal{F}(M_{iz})|^{2}$ (called intensity) which evolves very similarly to
the totally scattered intensity of our experiment (Fig. 2(a)). In particular,
the intensity increase is independent of the magnetic field in the first few
picoseconds and is later directly related to the density of skyrmions
nucleated in the final state.

In addition to providing a real-space picture of the magnetization textures
formed, the simulations also give access to the topological charge $Q$ of the
system. The evolution of $Q$ averaged over ten simulation runs is shown as
solid lines in Fig. 3(d). Already before the maximum temperature is reached,
topological charge is created and starts to fluctuate at all fields. The
period the system stays in the fluctuation state can be clearly identified by
rapid changes in the topological charge as exemplarily shown in Fig. 3(b). For
low magnetic fields, the topological charge stays constant after the
fluctuation state whereas, at high fields, $Q$ declines as does the intensity
$I$, both displaying the disappearance of topological spin textures. This
finding is again in line with our experimental observations.

To obtain a better understanding of the field dependence of the fluctuation
state, we performed another set of atomistic simulations where we let the
system relax to equilibrium at constant temperature either from a spin-aligned
ferromagnetic (FM) phase or from a random paramagnetic (PM) phase (SM section
III). We reproduce previous findings of an intermediate (IM) phase between the
FM and the PM phases at temperatures close to the Curie temperature
$T_{\text{C}}$ featuring skyrmions with a finite lifetime [14, 11].
Furthermore, we find that the topological charge $Q$ can be described
sufficiently accurately by the first-order approximation for the relaxation

$\frac{\mathrm{d}Q}{\mathrm{d}t}=\frac{Q_{\text{f}}-Q}{\tau_{\text{C}}},$ (1)

where $Q_{\text{f}}$ denotes the topological charge at equilibrium and
$\tau_{\text{C}}$ the relaxation timescale. Both $Q_{\text{f}}$ and
$\tau_{\text{C}}$ are dependent on the magnetic field $B$ and the temperature
$T$ and can be obtained from fits of the solution of Eq. (1) to the relaxation
data (see supplemental Fig. S8). Using temperature-interpolated data for
$Q_{\text{f}}$ and $\tau_{\text{C}}$, the basic rate equation in Eq. (1) can
be solved even for a (time-varying) temperature pulse to predict the
topological charge after the pulse at a given magnetic field. These
predictions, shown in Fig. 3(c) as dashed lines, qualitatively model the data
extracted from the atomistic simulations with a temperature pulse.

Looking closer at the relaxation time scale $\tau_{\text{C}}$, we find that it
diverges at low temperatures and low magnetic fields. At high temperatures,
i.e., in the IM phase, the relaxation time scales are of the order of
picoseconds for all fields, leading to a universal emergence of topological
charge at the peak of the temperature pulse. When the temperature drops and
the system is kept under low magnetic fields, the system cannot relax to the
equilibrium FM state and skyrmions effectively freeze out of the fluctuation
state due to the large relaxation time scales. At high fields, however,
$\tau_{\text{C}}$ is short enough for the system to relax to the FM state. The
topological switching is, therefore, a non-equilibrium phase transition that
happens on time scales that are much shorter than the relaxation time of the
system.

Figure 4: Energy barriers for skyrmion nucleation and decay extracted from
Arrhenius model. (a,b) Schematic of the field-dependent energy barriers (blue
curves): (a) at low temperatures skyrmion decay is only possible at high
fields; (b) at high temperatures skyrmions nucleate and decay at all fields
because the thermal fluctuations (red arrows) are large. (c), (d) Decay and
nucleation time scales $\tau_{\text{D}}$ and $\tau_{\text{N}}$, respectively,
as a function of the inverse temperature. Dashed lines represent Arrhenius
fits. (e) The energy barriers for the decay (circles, solid line) and the
nucleation (triangles, dotted line) determined from the Arrhenius fit as a
function of the magnetic field.

As the fluctuation state is characterized by rapid nucleation and decay of
topological charge [11], we now divide the relaxation time $\tau_{\text{C}}$
into two independent time scales: a nucleation time $\tau_{\text{N}}$ and a
decay time $\tau_{\text{D}}$ related to $\tau_{\text{C}}$ as
$1/\tau_{\text{C}}=1/\tau_{\text{N}}+1/\tau_{\text{D}}$. Since the balance of
nucleation and decay determines the equilibrium topological charge,
$Q_{\text{f}}$ is also a function of $\tau_{\text{N}}$ and $\tau_{\text{D}}$,
which allows us to find $\tau_{\text{N}}$ and $\tau_{\text{D}}$ from
$Q_{\text{f}}$ and $\tau_{\text{C}}$ as determined from the simulations (see
SM, section III, for complete derivation). We show the results for
$\tau_{\text{D}}$ and $\tau_{\text{N}}$ in Fig. 4(c) and (d), respectively,
and observe that both relaxation times are well described by an Arrhenius law
[13, 19, 20]. The resulting field-dependent energy barriers are shown in Fig.
4(e). The energy barrier for the skyrmion nucleation $E_{\text{N}}$ is almost
independent of the magnetic field, which explains why the fluctuation state
can robustly be accessed via photoexcitation independently of the applied
field as expected [17]. However, the energy barrier for the skyrmion decay
$E_{\text{D}}$ declines with the field and is almost zero for the highest
magnetic fields considered here. This explains why, at high magnetic fields,
the skyrmions do not freeze out like at lower fields. We illustrate the field-
dependent barriers and corresponding fluctuations at low and high temperature
in Fig. 4(a,b).

Interestingly, the energy barriers extracted from the relaxation simulations
have the same order of magnitude and show the same dependence on the magnetic
field as predictions for the energy barrier at zero temperature (SM, section
III) [21, 22, 20, 17, 23, 24]. This seemingly disproves the previous
suggestion that the fluctuation state is primarily initiated by a
substantially reduced energy barrier compared to the ground state [11]. In
fact, it is surprising that such high energy barriers are present in the high-
temperature, fluctuation-disordered phase [13, 14], where skyrmions have a
short lifetime. However, we note that the pre-exponential factor (attempt
frequency) is much higher than the commonly accepted micromagnetic estimate of
$\approx 1\text{\times}{10}^{9}\text{\,}\mathrm{H}\mathrm{z}$ (see
supplemental Fig. S11). Such substantially increased values for the attempt
frequency, including a strong dependence on the applied field over orders of
magnitude, were also found in previous calculations [20] and experiments [19]
investigating skyrmion stability and have been traced back to entropic
effects, which indeed differentiate between continuum and lattice models.
Interestingly, our results evidence that physical mechanisms elucidated before
at low temperatures (far from the Curie temperature) and on long timescales
(down to milliseconds) can also dictate topological fluctuation dynamics at
high temperature and on ultrafast timescales.

Nevertheless, we also emphasize that our atomistic simulations only feature a
simplified, minimal model for the skyrmion nucleation dynamics. In the
multilayer samples used in the experiments, the energy barrier is additionally
determined by the alignment between the ferromagnetic layers, which is absent
in our single-layer model. It is likely that eliminating the layer coupling
will lead to a reduction of the energy barriers at high temperatures,
promoting topological switching [17, 11]. While this effect would additionally
contribute to the high nucleation and decay rates in the fluctuation states,
it is insufficient to explain the freeze out of skyrmions when the temperature
drops again. In the ferromagnetic multilayer, the skyrmion state is stabilized
by stray fields even in the ground state. This long-range interaction is
missing in the atomistic simulations, and the frozen skyrmion state is
considered metastable. However, our rate model successfully explains the
experimentally confirmed, transient formation of skyrmions at applied fields
supporting a saturated ground state. Moreover, being based on only minimal
assumptions, the model suggests similar ultrafast laser-induced topological
switching and skyrmion formation mechanisms also for ferrimagnetic [25] and
antiferromagnetic [26] films with a small or vanishing net magnetization.

In conclusion, we here find that the fluctuation state can robustly be
accessed by ultrashort laser excitation. While the fluctuation state promotes
the topological switching into a skyrmions state, it also appears
independently of this transition (at high applied fields). We attribute this
robustness of the fluctuation state to a field-independent nucleation energy
barrier, whereas the topology of the final state is determined by diverging
relaxation time scales, resulting from a field-dependent skyrmion decay energy
barrier (Fig. 4(a,b)). Given the minimal assumptions in our model, the
reported asymmetric energy barrier for nucleation and decay indicates a
general mechanism for the creation of stable topological textures from non-
equilibrium states in other magnetic materials as discussed. This potentially
includes even systems with entirely different competing interactions, like
polar materials [27]. Further research on the fluctuation state will have to
address the role of long-range stray fields and interlayer coupling, which
ultimately may facilitate the discovery of the fundamental speed limit of the
topological transition in ferromagnetic multilayers and related materials.

## Acknowledgements

We acknowledge FERMI@Elettra Trieste for providing access to its free electron
laser facility. The authors thank Ingo Will from the Max Born Institute in
Berlin and Sebastian Wintz from the Max Planck Institute for Intelligent
Systems in Stuttgart for support with the measurements at MAXYMUS. This
research is part of the Shell NWO/FOM inititative “Computational sciences for
energy research” of Shell and Chemical Sciences, Earth and Life Sciences,
Physical Sciences, Stichting voor Fundamenteel Onderzoek der Materie (FOM) and
Stichting voor de Technische Wetenschappen (STW). Financial support from the
Leibniz Association via Grant No. K162/2018 (OptiSPIN), the Helmholtz Young
Investigator Group Program via Grant No. VH-NG-1520, and the European Research
Council under ERC Grant Agreement No. 856538 (3D-MAGiC) is acknowledged.

## Data Availability Statement

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

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