Current address: ]Max Planck Institut für Mikrostrukturphysik, Weinberg 2,
06120 Halle, Germany

# Collective topological spin dynamics in a correlated spin glass

Juraj Krempaský<EMAIL_ADDRESS>Photon Science Division, Paul Scherrer
Institut, CH-5232 Villigen, Switzerland Gunther Springholz  Institut für
Halbleiter-und Festkörperphysik, Johannes Kepler Universität, A-4040 Linz,
Austria Sunil Wilfred D’Souza  New Technologies-Research Center University of
West Bohemia, Plzeň, Czech Republic Ondřej Caha  Dept. of Condensed Matter
Physics, Masaryk University, Kotlářská 267/2, 61137 Brno ,Czech Republic
Martin Gmitra  Institute of Physics, P. J. Šafárik University in Košice, Park
Angelinum 9, 040 01 Košice, Slovakia Andreas Ney  Institut für Halbleiter-und
Festkörperphysik, Johannes Kepler Universität, A-4040 Linz, Austria Carlos
Antonio Fernandes Vaz  Photon Science Division, Paul Scherrer Institut,
CH-5232 Villigen, Switzerland Cinthia Piamonteze  Photon Science Division,
Paul Scherrer Institut, CH-5232 Villigen, Switzerland Mauro Fanciulli  LPMS,
Université de Cergy-Pontoise, 95031 Cergy-Pontoise, France Dominik Kriegner
Institute of Physics ASCR, v.v.i., Cukrovarnická 10, 162 53 Praha 6, Czech
Republic Dept. of Condensed Matter Physics, Charles University, Ke Karlovu 5,
121 16 Praha 2, Czech Republic Jonas A. Krieger  [  Laboratory for Muon Spin
Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Thomas Prokscha  Laboratory for Muon Spin Spectroscopy, Paul Scherrer
Institute, CH-5232 Villigen PSI, Switzerland Zaher Salman  Laboratory for Muon
Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland
Jan Minár<EMAIL_ADDRESS>New Technologies-Research Center University of
West Bohemia, Plzeň, Czech Republic J. Hugo Dil<EMAIL_ADDRESS>Photon
Science Division, Paul Scherrer Institut, CH-5232 Villigen, Switzerland
Institut de Physique, École Polytechnique Fédérale de Lausanne, CH-1015
Lausanne, Switzerland

###### Abstract

The interplay between spin-orbit interaction (SOI) and magnetic order is
currently one of the most active research fields in condensed matter physics
and leading the search for materials with novel and tunable magnetic and spin
properties. Here we report on a variety of unexpected and unique observations
in thin multiferroic Ge1-xMnxTe films. The ferrimagnetic order in this
ferroelectric semiconductor is found to reverse with current pulses six orders
of magnitude lower as for typical spin-orbit torque systems. Upon a switching
event, the magnetic order spreads coherently and collectively over macroscopic
distances through a correlated spin-glass state. Lastly, we present a novel
methodology to controllably harness this stochastic magnetization dynamics,
allowing us to detect spatiotemporal nucleation of topological spin textures
we term “skyrmiverres”.

Recently, combining semiconducting and magnetic properties within the same
material moved from being a concept DMS_book  to being one of the main
spintronic progress vectors Jungwirth_RMP_2014 , in particular when combining
magnetism with topological properties Dietl_RMP_2014 . With the advent of
ferroelectric Rashba semiconductors (FERS) Picozzi_Front , magnetic doping in
dilute magnetic semiconductors (DMS) has opened pathways to exploit the
electron spin associated with the Rashba-Zeeman type spin splitting of the
electronic states Fabian_RMP_2004 ; JK_GMT ; Yoshimi_GMT_2018 . To date the
primary FERS representative is $\alpha$-GeTe; with only two atoms per unit
cell and $\approx$0.3 Å displacement along the $\langle 111\rangle$ direction
between the Ge and Te atoms, it is arguably the simplest room temperature
ferroelectric semiconductorPawley_1966 ; JK_PRB . The combination of the
ferroelectric order and large SOI yields a switchable Rashba-type spin
structure of the bulk states JK_PRX , whereby the states are fully spin
polarised around the valence band maximum JK_PRR_2020 . When doped with Mn a
magnetic order is induced while the ferroelectric order remains at
temperatures up to 250 K for dopings below 25%, rendering it a multiferroic
semiconductor Springholz_PRL ; Kriegner_2016 ; JK_GMT . Moreover, the
collinear alignment of the magnetisation and FE polarisation axes ensures
magnetoelectric coupling in the system JK_PRX . Conversely, as there is now a
cohesive picture between Berry curvature and emergence of anomalous Hall
effect in non-centrosymmetric magnetic semiconductors Nagaosa_AHE_RMP_2010 ,
the current-induced magnetization switching mechanism in this material points
to a bulk Rashba-Edelstein effect Yoshimi_GMT_2018 . Both spectroscopic JK_GMT
; JK_PRX  and transport studies Yoshimi_GMT_2018  indicate that the key
ingredient for Ge1-xMnxTe magnetoelectric functionality is its itinerant
magnetism mediated by the valence band Fukuma_2005 ; Sato_JPRPh_2005 ;
Fukuma_2006 ; Fukuma_2008 ; Fukushima_2014 . Furthermore, this material is
proposed to be a unique platform to explore novel phenomena such as
nonreciprocal electric transport Yoshimi_PRB  or magnetic skyrmionic polarons
Brey_Nano_2018 .

By combining a range of spectroscopic techniques based on X-ray magnetic
circular dichroism (XMCD), near-edge x-ray absorption fluorescence
spectroscopy (NEXAFS) and state of the art modeling (see Methods I,II,III,IV),
we show that Ge0.87Mn0.13Te is a bulk ferrimagnet (FiM). These results are
further expanded upon with complementary low-energy muon spectroscopy
(LE-$\mu$SR) as a local probe to characterize temperature-dependent magnetic
transitions on nanometers scale; and SQUID magnetometry used to investigate
the macroscopic magnetisation. The results testify that the FiM order builds
up below $T_{c1}=100$ K on top of a paramagnetic (PM) background and that
below $T_{c2}=40$ K the system behaves as a spin-glass. Developing new
analysis tools based on magnetostochastic resonance (MSR) switching allows us
to characterize the Ge0.87Mn0.13Te magnetism in static and dynamical regimes.
The ensemble of experimental techniques and theory yields a holistic view of a
correlated spin glass with emergent topological spin textures. Our data
indicate that XMCD can detect these topological spin-textures, that
magnetization switching can be driven by nanoampere pulsed currents using
stochastic resonance, that this switching is collective due to glassiness, and
that the magnetic switch can be turned on and off by small changes in pulse
periodicity.

### Ferrimagnetic full switching

Figure 1a shows XMCD spectra measured at 10 K in total electron yield (TEY)
mode while applying selected magnetic fields up to $\pm$ 6 Tesla. As will be
discussed later, the dichroism signal does not saturate at our maximum
B-fields due to a paramagnetic spin-glass state, characteristic also for
Cd1-xMnxTe related DMS system Galazka_1980 . The top-inset is a zoom-in into
the $\mathrm{Mn_{L3}}$ absorption edge, showing two distinct XMCD features
separated by $\approx$0.5 eV. A time series of XMCD spectra for selected
B-fields in Fig. 1b shows that these features are fluctuating, eventually
completely and spontaneously reversing at zero B-field (Fig. 1c). The shape of
the dichroism signal at the $\mathrm{Mn_{L3}}$-edge can be understood from the
combination of a paramagnetic background superposed with ferrimagnetic
contributions (1d). The fluctuating spectral weight in Fig. 1b indicates that
the switching continues for applied fields even above 4 T. From considering an
extensive data set obtained from several experimental runs and samples, the
magnetic reversal appears to be regular and does not show the telegraph noise
expected for a system which randomly switches between two magnetic, or spin,
states Natterer_PRL_2018 ; Hermenau_NC2019 . As surprising as the switching
itself, is the fact that it occurs simultaneously, within the time scale of
the experiment, over at least the square millimetre area of the photon spot.

To reiterate, the dichroism effect at $\mathrm{Mn_{L3}}$ systematically shows
two opposite magnetic moments from different types of Mn atoms, with
absorptions at $h\nu=$638.8 and 639.3 eV, simultaneously switching after about
every energy scan across the $\mathrm{Mn_{L3}}$ absorption edge, in absence or
presence of an externally applied field. Notably, the opposite magnetic
moments are not exactly equal in magnitude, indicating a non-balanced
ferrimagnetic order from two distinct Mn-sites. The spontaneous switching is
further corroborated by the fact that we obtain equivalent dichroism effect by
measuring spectra with the same circular polarization (see Extended Fig. 1a).
This also indicates that the variation of light helicity is not the driving
force of the switching. There are now three main question which arise:

  * •

What is the difference between the two Mn-sites?

  * •

Why are their magnetic moments spontaneously reversing in time over
macroscopic length scales?

  * •

How to control the switching?

We first identify the different Mn contributions with our theoretical
calculations, and proceed with the magnetostochastic resonance switching model
to address the latter two questions.

### Magnetic ground state properties

In accordance with our earlier resonant ARPES studies JK_GMT  and extensive
XMCD studies on other Mn-doped DMS systems GvL_PRB_GaMnN_2007 ; GvL_2014 ;
GvL_GaMnAs_2015 , we identify these moments to originate from substitutional
($\mathrm{Mn_{s}}$) and interstitial ($\mathrm{Mn_{i}}$) atoms, respectively.
However, in contrast to related systems we find that in Ge0.87Mn0.13Te the
characteristic $\approx$0.5 eV energy separation between $\mathrm{Mn_{s}}$ and
$\mathrm{Mn_{i}}$ is not related to Mn segregation effects, or to non-magnetic
surface Mn-oxides, nor to a depletion of Mn in the near-surface layer.
Instead, the energy shift is an intrinsic bulk property we also see in bulk-
sensitive fluorescence yield and in NEXAFS analysis from the Mn-K edge (see
Extended Fig. 1b-c, respectively). This confirms the presence of
$\mathrm{Mn_{s}}$ and $\mathrm{Mn_{i}}$ atoms, with a
$\mathrm{Mn_{s}}$:$\mathrm{Mn_{i}}$ occupancy close to 2:1. We employ this
experimental finding to calculate the Ge0.87Mn0.13Te magnetic ground state
properties using density functional theory, where the Mn-disorder was
implemented within the coherent potential approximation (CPA) alloy theory
(see Methods III).

Figure 2 summarizes that Mn-doping induces a frustrated magnetism with various
exchange interactions, while keeping the polar crystal structure. For 13% Mn-
doping we expect nearly 1 Mn per 8 unit cells, or 1 Mn atom randomly
distributed over 4 unit cells in 2D. Based on the CPA formalism, Fig. 2a
illustrates the Mn-occupancy in an “infinite” Ge0.87Mn0.13Te lattice
consisting of 91.1% probability occupation of 8.9% $\mathrm{Mn_{s}}$
substituted Ge ($\mathrm{Mn_{1}}$), and two types of interstitial
$\mathrm{Mn_{i}}$ atoms $\mathrm{Mn_{2}}$ and $\mathrm{Mn_{3}}$, each one with
2.2% probability occupation. The most prominent contribution to the magnetic
order comes from the Heisenberg exchange energy between $\mathrm{Mn_{2,3}}$
and between $\mathrm{Mn_{1,2/3}}$ atoms, whereas the interaction between
substitutional atoms ($\mathrm{Mn_{1,1}}$) becomes damped due to the presence
of the interstitial Mn (Fig. 2b). The corresponding magnetic exchange
couplings $J_{i,j}$ are presented as a function of the distance $R_{i,j}/a$
between atoms, where $a$ is the lattice parameter and $i,j=1,2,3$. The
positive exchange constants ($J_{i,j}>0$) favours ferromagnetic (FM) order,
whereas negative values ($J_{i,j}<0$) favour antiferromagnetic (AFM) order
(see Methods IV).

The co-presence of FM/AFM in a DMS system are the main ingredients for
magnetic frustration in a canonical spin glass, which in our case is further
assited by Dzyaloshinskii-Moriya interaction (DMI). DMI is characteristic to
non-centrosymmetric systems with large spin-orbit coupling (SOC) such as
Ge1-xMnxTe, where it promotes non-coplanar (canted) arrangement between spin
states summarized in Fig. 2c. Both DMI and all $J_{i,j}$ have fluctuating
$R_{i,j}/a$ dependence. Furthermore the DMI is non-negligible compared to the
$J_{i,j}$, especially for larger distances, indicating significant magnetic
frustration. For small length scales there is a competition between FM and AFM
contributions, the relative magnitudes of which explain why Ge0.87Mn0.13Te
stabilizes in a ferrimagnetic ground state. Since the CPA approach allows us
to address spectral contributions from individual Mn-atoms (Fig. 2d), the
resulting theoretical magnetic dichroism can reproduce the measured spectra
very well if we take into account that the majority of $\mathrm{Mn_{s}}$
contribute to the PM background once the magnetic interactions stabilize, as
detailed in Extended Fig. 2d.

The here described FiM order, based on three distinct interacting Mn lattice
sites as the ground state of the system, was elusive in previous theoretical
Antonov_GMT_CMP_2010 ; Fukushima_2014  and experimental studies Fukuma_2005 ;
Fukuma_2006 . The main reason for this is that in these studies only
substitutional $\mathrm{Mn_{s}}$-sites were considered (empty markers in Fig.
2b), whereas in our case their oscillatory long-range $J_{i,j}$ exchange is
damped due to the presence of interstitial $\mathrm{Mn_{i}}$ atoms (filled
green markers in Fig. 2b). Furthermore, when probed under applied B-field
(Fig. 1b), the opposite FiM contributions are placed on a large PM background
due to which the $\mathrm{Mn_{i}}$-moments partially align with those of
$\mathrm{Mn_{s}}$. Only the dynamics shown in Fig. 1c allows to distinguish
the two sites under applied magnetic field.

In order to assess the energetics of the FiM switching, Fig. 3a compares the
energy scale of the rotation of $\mathrm{Mn_{i}}$ spins with respect to the
$\mathrm{Mn_{s}}$ spin held fixed along the [111] axis with the
magnetocrystalline anisotropy energy (MAE, see Methods IV). The combination of
both energy profiles constitutes a double-well uniaxial potential with MAE
barrier of $\approx$0.1 meV comparable to ferromagnetic $3d$ metals MAE_PRL_95
. Having identified the general magnetic order and its bistable energy
potential model, we now turn to the dynamics and the unexpected switching.

### Magnetostochastic resonance switching

Generally speaking, stochastic resonance (SR) is a phenomenon applicable to
nonlinear systems whereby a weak signal is amplified through an entraining
periodic signal or even noise. Besides the bistable potential, SR requires two
additional ingredients Gammaitoni_RMP_1998 : (i) a weak periodic input and
(ii) a source of noise entrained with the periodic input. As will be
elaborated below, the first ingredient is provided by periodic TEY
oscillations upon Ge0.87Mn0.13Te x-ray illumination, whereas the second
originates from the short TEY pulse in x-ray absorption during the scan over
the $\mathrm{Mn_{L3}}$ absorption edge. The sum of these assists the
transition between the two FiM equilibria as depicted in Fig. 3a,b.

The periodic drive $P(t)$ is in our case best described by a continuous
sinusoidal signal with $T_{\Omega}\approx 5$ minutes (Fig. 3c). These
systematically observed oscillations are present also while performing h$\nu$
scans; they are not related to the periodic top-up current refill of the
synchrotron and do not depend on x-ray polarization and applied B-field (see
Extended Fig. 3a-c). Therefore, our understanding is that they originate from
charging and discharging inside a capacitor circuit with Ge0.87Mn0.13Te as
dielectric, whereby the primary charging effect is caused by the ferroelectric
order which under x-rays produce a charge separation due to a steady-state
bulk photovoltaic effect vonBaltz_1981 . For the host GeTe this effect is even
stronger compared to conventional oxide ferroelectrics Rappe_GeTeSHC_PRL_2018
. Moreover, the system possesses self-poling properties from host GeTe
JK_MDPI_2019 , and negative capacitance Orlova_negGeTeC_arXiv , leading to
thermodynamically unstable transient effects Hoffmann_APL_2021 . Finally, the
TEY current, especially as a short pulse $\eta(t)$ at the
$\mathrm{Mn_{L3}}$-edge (Fig. 3c), induces spin-orbit torque via the above
mentioned Rashba-Edelstein effect Yoshimi_GMT_2018 .

A phenomenological description of magnetostochastic resonance (MSR) predicts a
million times higher switching probability for bistable magnetic systems
Grigorenko_IEEE_1995 . Consistent with these predictions, for Ge0.87Mn0.13Te
the current-driven magnetisation switching in the milliampere regime
Yoshimi_GMT_2018  is achieved under MSR with a nanoampare TEY pulse at the
$\mathrm{Mn_{L3}}$ absorption edge. This is implemented with on-the-fly
absorption scans JK_OTF_AIP_2010 , in contrast to conventional slow point-by-
point scans, where the FiM switching is not observed. A typical measurement
series is denoted by the green top trace in Fig. 3d, where the resulting TEY
signal is a two-frequency system in which $P(t)$ realizes frequency modulation
and $\eta(t)$ implements amplitude modulation Song_AMFM_SR_2016 . With such a
combination of driving frequencies the SR can be harnessed, e.g. controllably
suppressed by detuning the frequencies from their optimal setting. We achieved
this effect by measuring XMCD data with different energy ranges (grey bottom
trace in 3d), thereby moving away from the optimal synchronization with
driving frequency $t_{1}=t_{2}=\frac{1}{2}t_{\Omega}$ Gammaitoni_RMP_1998
where $t_{1,2}$ is the time needed to measured one XAS spectrum, and
$t_{\Omega}$ one XMCD dataset.

Figure 4 summarizes the MSR harnessing concept with two synchronized loops
with aperiodic (MSR-off) and periodic additive noise (MSR-on). Data indicate
that there is an immediate change from the MSR-off regime without FiM
switching to FiM switching under the MSR-on regime.

We emphasize that under MSR-on the FiM switching holds only in the statistical
average, quantified via the residence time distribution $N(t)$
Gammaitoni_RMP_1998 . We quantified $N(t)$ by counting the multiples of
$t_{\Omega}$ between subsequent FiM switching occurrences, leading to a
sequence of Gaussian-like peaks with exponentially decreasing envelope as seen
in Fig. 4d at 5 K. In qualitative agreement with SR predictions
Gammaitoni_RMP_1998 , the peak distributions under thermal noise shifts to
shorter times, as seen at 10 K. To further confirm the SR origin of FiM
switching, the $N(t)$ distribution envelope is found to depend on the
resonance conditions through the driving period $t_{\Omega}$ (see Extended
Fig. 4a).

### Topological spin textures

In order to obtain a microscopic understanding of the MSR switching, we
analyze the orbital $\mathrm{m_{L}}$ and spin $\mathrm{m_{S}}$ contribution
from sum rules Thole_PRL_92 ; Carra_PRL_93 . Because the magnitude and sign of
the MSR-on XMCD signal changes in time, the directional sense of the magnetic
moments with respect to the beam axis is changing as well, occasionally
measuring a totally quenched moment
$\bf{M}_{tot}=\mu_{B}(2\mathrm{m_{S}}+\mathrm{m_{L}})$ such as in the grey
traces in Fig. 4c. The B-field dependence of $\mathrm{m_{L}}$ and
$\mathrm{m_{S}}$ contributions obtained for MSR-off and MSR-on regimes is
summarized in Fig. 5. For Mn the theoretically obtained $\mathrm{m_{L}}$ of
-0.032$\mu_{B}$/(Mn atom) is independent of the magnitude of the applied field
and the total magnetisation $\bf{M}_{tot}$ is expected to be dominated by
$\mathrm{m_{S}}$, which takes a value of
$\mathrm{m_{S}}$$\approx$2.1$\mu_{B}$/(Mn atom) at saturation. This trend is
well observed in Fig. 5a where in the MSR-off regime the applied B-field
drives $\mathrm{m_{S}}$ whereas $\mathrm{m_{L}}$ is rather constant. That the
local Mn-moments deduced from the sum rules (0.17$\pm$0.01 $\mu_{B}$/Mn atom)
for B=0 differ from the theoretical values is due to the fact that most Mn-
moments are canted away from the surface normal. Table 2 in Methods III
compares the calculated Mn-moments with experimental data under MSR-off/on
regimes.

The MSR-on data in Fig. 5b,c show a similar picture with $\mathrm{m_{S}}$
following the magnetic field strength and $\mathrm{m_{L}}$ initially small and
nearly constant. However, when the field direction is reversed, the
$\mathrm{m_{L}}$ starts to strongly deviate from this constant. If the field
continues to be rapidly changed, i.e. after each XMCD data pair, the
$\mathrm{m_{L}}$ values return to the baseline (Fig. 5b). However, as shown in
Fig. 5c, if the change in B is slower and the field steps smaller, the orbital
moment $\mathrm{m_{L}}$ appears to ring up acquiring values that rise far
above meaningful values for Mn atoms in our system. Moreover, the spin
contribution $\mathrm{m_{S}}$ appears to be dragged along and increases to
values expected for higher applied B-field, together leading to an
exponentially increasing oscillatory total moment (Fig. 5c inset). This
behavior and these unexpected high values strongly indicate a collective
effect and imply a nucleation of topological spin textures (spin vortices)
during the field sweeping, reminiscent to skyrmion nucleation
dosSantos_NC_2016 . The dependency on how fast the field is changed indicates
that the “spin viscosity” changes, in correspondence with the spin glass
character discussed below.

A macroscopic perspective on skyrmion nucleation in Ge0.87Mn0.13Te is attested
by SQUID magnetometry (Fig. 5d). First, the presence of topological spin
textures in such films were anticipated from non-monotonic decrease (dips) in
the magnetic field dependence of the anomalous Hall component Yoshimi_GMT_2018
. For bulk ferromagnets lacking inversion symmetry such dips are related to
competition between the metastable skyrmions aggregation and magnetic domains,
resulting in a non-equilibrium state under varying B-field
Tokura_FeGe_NPhys_2018 . Because the remanent magnetization is lower for
Ge0.87Mn0.13Te samples grown on InP as for those grown on BaF2, this re-
aggregation effect is more pronounced in the former (Fig. 5d). Thus the dips
indicated by arrows are indirect evidence of metastable skyrmion re-
aggregation Tokura_FeGe_NPhys_2018 . Finally, the unusual hysteresis shape
with open area and narrow waist is characteristic to Néel-type skyrmions
hosted in perpendicular magnetic thin-film ferromagnets Liu_ChPL_2022 ;
Buttner_NM_2021 . In our XMCD experiments the sudden $\mathrm{m_{L}}$ changes
upon magnetic field reversal serve as a direct probe for the emergence of
topological spin-textures, highlighting that XMCD can indeed detect
unconventional magnetic structures such as skyrmions dosSantos_NC_2016 .

### Correlated spin glass

Already from the shape of the M(T) magnetization curve in Fig. 6b it is
evident that the system does not behave like a ferromagnet with typical
magnetisation $(1-(T/T_{c})^{\alpha})^{\beta}$ dependence and a single
transition temperature, but rather a steadily diminishing magnetization with
various critical temperatures indicated in the figure. The splitting between
zero-field-cooled (ZFC) and field-cooled (FC) curves is a clear indication of
competing energy scales and spin orders, and of possible spin excitations in
magnetic glassy-phases Vincent_SG_2018 . LE-$\mu$SR is the method of choice to
characterize magnetically ordered volumes inside such a spin-glass from a
local perspective. Fig. 6a summarizes three quantities extracted from our
measurements: (i) the local mean field $B$ sensed by the implanted muons
(top); (ii) the damping rate $\lambda$ of the oscillations which is
proportional to the width of local field distribution (middle); and (iii) the
initial amplitude of the precessing muon spin polarization $A_{0}$ (bottom).
All three parameters start to change around the onset of magnetic order
($\approx$100 K) and show either a minimum ($B$), maximum ($\lambda$), or
plateau ($A_{0}$) at 40 K. The combination of bulk magnetization and
LE-$\mu$SR data provides clear evidence that the magnetic order has a net
magnetic moment imposed by the FiM order below $T_{c1}\approx$100 K, followed
by a spin-glass transition below $T_{c2}\approx$40 K where the coupling
between the FiM clusters becomes strong enough for the sample to undergo a
transition to a cluster spin-glass long range order.

The combined set of experimental and theoretical results allows us to
rationalize the details of the MSR switching mechanism. For a broader
perspective Fig. 6b compares the temperature dependence of $\mu$SR local mean
field, bulk magnetization, and XMCD switching statistics. As expected, the
spin-freezing temperature $T_{c2}$ is marking a minimum rather than an onset
in FiM switching probability. Strikingly, as highlighted by the red area,
above $T_{c2}$, the switching events dramatically increase along with the
decreasing size of the FiM clusters; and this trend appears to be independent
on whether the system was magnetized under ZFC or field-heated field-cooled
(FH-FC) indicated by green and red markers, respectively. Moreover, at T=65 K
the XMCD asymmetry sometimes reaches values as high as 60% in contrast to the
expected increase of magnetic disorder at elevated temperatures (Extended Fig.
4). This is reminiscent of the spontaneous thermally induced magnetic order in
elemental spin glasses Verlhac_NP_2022 , which in our case begins to morph
between individual FiM clusters as indicated by pictographs in Extended Fig.
4b. We expect that such morphing/clustering create new spin vortex patterns
which also affect the PM background, thereby enhancing the dichroism effect
close to theoretical predictions in Fig. 2b. That kind of reordering on a
mesoscopic scale is extremely slow, explaining the “spin viscosity” dynamical
behavior seen in Fig. 5c inset.

The slow, viscous, reordering on a mesoscopic scale (Fig. 5c inset) and the
magnetic frustration highlighted in the theory section, are directly related
to the glassy behavior. Furthermore, our XMCD results show that the FiM
switching is collective over macroscopic dimensions. This indicates that the
magnetic order has to be correlated across local magnetic interactions on
which the magnetic spin textures spread like a collective excitation through
the sample with spatiotemporal net spin windings generating the oscillatory
magnetism as indicated in Fig. 5c inset. This is exactly what is expected for
a correlated cluster spin-glass Ochoa_SHD_PRB ;
EManiv_collective_AFM_Science_2021  which under certain applied B-fields,
attained after proper initialisation with B-field sweeping, can nucleate
topological spin vortices on a mesoscopic scale.

## Conclusions

Our Ge0.87Mn0.13Te thin films present a variety of intriguing physical
phenomena which we could decipher using our holistic approach; XMCD and theory
confirm a bistable ferrimagnetic ground state, whereas the collective
switching of this state in XMCD, together with $\mu$SR and magnetometry,
indicate a cluster spin glass, and SQUID and XMCD show the emergence of
topological spin textures supported by the large DMI found in theory. To
highlight the novel concept of spatiotemporal topological spin textures
arising from a spin glass, we name them “skyrmiverres”. Given the rich phase
diagram of most spin glasses with respect to variation of external parameters,
we expect that such skyrmiverres under dynamical conditions are ubiquitous in
multiferroic semiconductors such as Ge0.87Mn0.13Te with metastable topological
spin states coupled to magnetic and ferroelectric order. It will be of
particular interest to determine the shape and distribution of these
skyrmiverres using nanoscopic imaging techniques, especially in combination
with pulsed currents to follow their switching behavior. For bistable systems
we established a concept to harness magnetostochastic switching in XMCD
experiments which allowed us to describe novel magnetization dynamics. Under
this MSR drive the current needed for switching of the magnetic state is
reduced by six orders of magnitude, allowing for the design of energy
efficient non volatile FiM electronics possibly employing skyrmiverres as
information carriers. Lastly, the combination of volatile and non-volatile
elements and long and short term memory could make the system a promising
playground for neuromorphic memory. On a more fundamental level, our results
open up new avenues for studying collective topological spin dynamics in
correlated spin glass systems.

## References

  * (1) J. Kossut, J. A. Gaj, eds., Introduction to the Physics of Diluted Magnetic Semiconductors (Springer-Verlag Berlin Heidelberg, 2011).
  * (2) T. Jungwirth, et al., Rev. Mod. Phys. 86, 855 (2014).
  * (3) T. Dietl, H. Ohno, Rev. Mod. Phys. 86, 187 (2014).
  * (4) S. Picozzi, Frontiers in Physics 2 (2014).
  * (5) I. Zutić, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
  * (6) J. Krempaský, et al., Nat. Commun. 7, 13071 (2016).
  * (7) R. Yoshimi, et al., Science Advances 4, eaat9989 (2018).
  * (8) G. S. Pawley, W. Cochran, R. A. Cowley, G. Dolling, Phys. Rev. Lett. 17, 753 (1966).
  * (9) J. Krempaský, et al., Phys. Rev. B 94, 205111 (2016).
  * (10) J. Krempaský, et al., Phys. Rev. X 8, 021067 (2018).
  * (11) J. Krempaský, et al., Phys. Rev. Research 2, 013107 (2020).
  * (12) H. Przybylińska, et al., Phys. Rev. Lett. 112, 047202 (2014).
  * (13) D. Kriegner, et al., Phys. Rev. B 94, 054112 (2016).
  * (14) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
  * (15) Y. Fukuma, et al., AIP Conference Proceedings 772, 349 (2005).
  * (16) H. Sato, et al., Journal of Electron Spectroscopy and Related Phenomena 144-147, 727 (2005). Proceeding of the Fourteenth International Conference on Vacuum Ultraviolet Radiation Physics.
  * (17) Y. Fukuma, et al., Journal of Applied Physics 99, 08D510 (2006).
  * (18) Y. Fukuma, et al., Applied Physics Letters 94, 269901 (2009).
  * (19) T. Fukushima, et al., Journal of Physics: Condensed Matter 27, 015501 (2014).
  * (20) R. Yoshimi, et al., Phys. Rev. B 106, 115202 (2022).
  * (21) L. Brey, Nano Letters 17, 7358 (2017).
  * (22) R. R. Galazka, S. Nagata, P. H. Keesom, Phys. Rev. B 22, 3344 (1980).
  * (23) F. D. Natterer, F. Donati, F. m. c. Patthey, H. Brune, Phys. Rev. Lett. 121, 027201 (2018).
  * (24) J. Hermenau, et al., Nature Communications 10, 2565 (2019).
  * (25) A. A. Freeman, et al., Phys. Rev. B 76, 081201 (2007).
  * (26) G. van der Laan, A. I. Figueroa, Coordination Chemistry Reviews 277-278, 95 (2014). Following Chemical Structures using Synchrotron Radiation.
  * (27) K. Edmonds, G. van der Laan, G. Panaccione, Semiconductor Science and Technology 30, 043001 (2015).
  * (28) V. Antonov, A. Shpak, L. Bekenov, L. Germash, A. Yaresko, Condensed Matter Physics 13, 1 (2010).
  * (29) J. Trygg, B. Johansson, O. Eriksson, J. M. Wills, Phys. Rev. Lett. 75, 2871 (1995).
  * (30) L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).
  * (31) R. von Baltz, W. Kraut, Phys. Rev. B 23, 5590 (1981).
  * (32) S.-J. Gong, F. Zheng, A. M. Rappe, Phys. Rev. Lett. 121, 017402 (2018).
  * (33) D. Kriegner, et al., Crystals 9, 335 (2019).
  * (34) N. N. Orlova, A. V. Timonina, N. N. Kolesnikov, E. V. Deviatov, Dynamic negative capacitance regime in GeTe Rashba ferroelectric, arXiv DOI: 10.48550/ARXIV.2206.00346 (2022).
  * (35) M. Hoffmann, S. Slesazeck, T. Mikolajick, APL Materials 9, 020902 (2021).
  * (36) A. Grigorenko, P. Nikitin, IEEE Transactions on Magnetics 31, 2491 (1995).
  * (37) J. Krempaský, et al., AIP Conference Proceedings 1234, 705 (2010).
  * (38) H. Song, S. Huang, X. Liu, International Journal of Modern Physics B 30, 1650113 (2016).
  * (39) B. T. Thole, P. Carra, F. Sette, G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992).
  * (40) P. Carra, B. T. Thole, M. Altarelli, X. Wang, Phys. Rev. Lett. 70, 694 (1993).
  * (41) M. dos Santos Dias, J. Bouaziz, M. Bouhassoune, S. Blügel, S. Lounis, Nature Communications 7, 13613 (2016).
  * (42) X. Yu, et al., Nature Physics 14, 832 (2018).
  * (43) J. Liu, et al., Chin. Phys. Lett. 39, 017501 (2022).
  * (44) F. Büttner, et al., Nature Materials 20, 30 (2021).
  * (45) E. Vincent, V. Dupuis, Spin Glasses: Experimental Signatures and Salient Outcomes (Springer International Publishing, Cham, 2018), pp. 31–56.
  * (46) B. Verlhac, et al., Nature Physics 18, 905 (2022).
  * (47) H. Ochoa, R. Zarzuela, Y. Tserkovnyak, Phys. Rev. B 98, 054424 (2018).
  * (48) E. Maniv, et al., Science Advances 7, eabd8452 (2021).
  * (49) The FiM switching statistical error margins are binomial confidence intervals (obtaining $x$-switches in $n$ trials gives a $\sqrt{(}x/n*(1-x/n)/n)$ error). Statistically most relevant FiM switching summarized from three independent beamtimes from fresh samples was measured at 5 K (ZFC), that is why the smallest error bar in FiM switching probability in Fig. 6b is achieved under these measurement conditions.
  * (50) C. Piamonteze, et al., Journal of Synchrotron Radiation 19, 661 (2012).
  * (51) T. Prokscha, et al., Nuc. Inst. Phys. A 595, 317 (2008).
  * (52) A. Suter, B. Wojek, Physics Procedia 30, 69 (2012).
  * (53) H. Ebert, D. Ködderitzsch, J. Minár, Rep. Prog. Phys. 74 (2011).
  * (54) S. H. Vosko, L. Wilk, M. Nusair, Canadian Journal of Physics 58, 1200 (1980).
  * (55) P. Soven, Phys. Rev. 156, 809 (1967).
  * (56) D. W. Taylor, Phys. Rev. 156, 1017 (1967).
  * (57) W. Johnston, D. Sestrich, Journal of Inorganic and Nuclear Chemistry 19, 229 (1961).
  * (58) E. Goering, Philosophical Magazine 85, 2895 (2005).
  * (59) A. Liechtenstein, M. Katsnelson, V. Antropov, V. Gubanov, Journal of Magnetism and Magnetic Materials 67, 65 (1987).
  * (60) B. Belhadji, et al., Journal of Physics: Condensed Matter 19, 436227 (2007).

## Acknowledgments

We thank Stefano Rusponi and Valerio Scagnoli for fruitful discussions. We
gratefully acknowledge financial support from the Swiss National Science
Foundation project No. PP00P2_170591. Ja.M. and SdS would like to thank the
CEDAMNF (Grant No. CZ.02.1.01/0.0/0.0/15_003/0000358) co-funded by the
Ministry of Education, Youth and Sports of Czech Republic and the GACR Project
No. 20-18725S for funding. G.S. acknowledges support by the Austrian Science
Funds (FWF), Projects P30960-N27 and I4493-N. DK acknowledges the Lumina
Quaeruntur fellowship of the Czech Academy of Sciences. All experimental data
will be made available upon request.

Figure 1: Switching of ferrimagnetic order in XMCD. (a) XMCD spectra for
selected applied $B$-fields. (Top inset) zoom into the Mn $L_{3}$ edge,
showing two distinct spectral weight from substitutional $\mathrm{Mn_{s}}$ and
interstitial $\mathrm{Mn_{i}}$ atoms. (Bottom inset) background subtracted XAS
spectra for $\pm\mu$ light polarizations, measured at B=6 T (10 K), and
overlaid with multiplet calulations. (b) Series of $\mathrm{Mn_{L3}}$ XMCD
spectra taken at $\approx$3 min intervals for selected B-fields. (c)
Spontaneous switching recorded in two consecutive XMCD datasets (red/blue) in
the absence of applied B-field, shown at the $\mathrm{Mn_{L3}}$ and
$\mathrm{Mn_{L2}}$ absorption edges. d) $\mathrm{Mn_{L3}}$-edge model based on
4 Gaussian functions simulating $\mathrm{Mn_{s}}$ and $\mathrm{Mn_{i}}$
spectral contributions. For PM the $\mathrm{Mn_{s}}$ and $\mathrm{Mn_{i}}$
features are aligned, with their amplitude dependent on the B-field. The FiM
consists of oppositely aligned peaks with different amplitude. $M$ and
$M_{SW}$ refer to the reference and switched state respectively.  Figure 2:
Frustrated magnetic interactions in Ge0.87Mn0.13Te. (a) Illustration of Mn
doping in the rhombohedrally distorted $\alpha$-GeTe unit cell with one
substitutional $\mathrm{Mn_{s}}$ and two potential interstitial
$\mathrm{Mn_{i}}$ dopants. (b) Exchange couplings $J_{i,j}$, between
$\mathrm{Mn_{i}}$-$\mathrm{Mn_{j}}$ atoms with all three types of Mn atoms
present (full markers), and with only substitutional $\mathrm{Mn_{1}}$ atoms
(empty marker). Distances are normalized to the lattice parameter a. (Inset)
Extension to smaller atomic distances. (c) DMI values for inter- and intra-Mn
site coupling. The $D_{i,j,x}$ component is summarized as $xi,j$ and similar
for $y$ and $z$ ($x,y,z$ index directions are denoted in a). Dominant
interactions are indicated in the graph, others listed in legends. (d)
Comparison between XMCD experiment (red dotted line) and CPA theory, in which
spectral contributions from individual Mn atoms result in a final XMCD signal
(dashed line). Figure 3: Magnetostochastic resonance driving sources. (a)
Stochastic resonance switching on a symmetric double-well qualitatively
modelled by FiM exchange energy (grey markers) and magnetocrystalline
anisotropy energy (red markers). The potential is subject to a synchronised
periodic drive $\mathrm{T_{\Omega}}$ (red-yellow trace), enabling the
switching of the otherwise stable FiM state. (b) FiM switching cartoon: the
circles illustrate the MSR harnessing concept with periodic $\mathrm{Mn_{L3}}$
TEY pulses indicated by grey and red/blue flash symbols, respectively.
Periodic pulses with entrained $P(t)$ signal (red-yellow circle) enable the
FiM switching in a statistical average. (c) Sinusoidal fit (red-yellow trace)
of a time evolution of normalized TEY periodic drive $P(t)$ measured at 10 K,
B=0, and $h\nu=638.8$eV (grey trace). Superimposed red and blue spikes
represent transient TEY pulse at $\mathrm{Mn_{L3}}$ edge that would occur
during XAS energy scans. (d) Typical time series of periodic (green/top) and
aperiodic (grey/bottom) XAS scans which controls the magnetostochastic
resonance switching (MSR-on/off). Empty spaces between the XAS scans is time
needed to move the monochromator back to the initial energy. Figure 4:
Harnessing switching dynamics with magnetostochastic resonance. (a) Stacked
plot of $\mathrm{Mn_{L3}}$ XMCD spectra measured in MSR-off regime directly
followed by MSR-on regime. (b) XMCD MSR-off spectra averaged for individual
applied B-fields as depicted in the inset. Top cartoon illustrates that under
aperiodic $\mathrm{Mn_{L3}}$ TEY pulses the MSR switching is disabled. (c)
Stacked plot of MSR-on XMCD spectra. In (a) and (c) blue/red traces reflect
the original or switched FiM state, grey traces reflect totally quenched
magnetic state. (d) Statistical FiM switching quantification by residence time
distributions $N(t)$, fitted with Gaussian functions at 5 and 10 K. Figure 5:
Emergence of topological spin textures in XMCD and SQUID. (a-c) Time series of
spin ($\mathrm{m_{S}}$) and orbital ($\mathrm{m_{L}}$) moments showing their
dependence on applied B-field under MSR-off (panel a) and MSR-on regimes
measured with “fast” (panel b) and “slow” B-field step changes (panel c). Data
in (a) are measured after zero-field cooling (ZFC), data in (b-c) after full
warming/cooling cycle under applied field of 6 Tesla (FH-FC). The inset in c
shows the total moment time dependence $\mathrm{M_{tot}}$ from data inside the
dashed frame. (d) Out-of-plane SQUID hysteresis for Ge0.87Mn0.13Te grown on
InP(111) (blue markers, right inset), and $\mathrm{BaF_{2}}$(111) substrates
(left inset). Figure 6: Correlated spin glass state observed by LE-$\mu$SR,
SQUID, and switching dynamics. (a) Temperature-dependent average local field
$B$ (top), muon damping rate $\lambda$ (middle), the initial amplitude $A_{0}$
(bottom), obtained from fits of the LE-$\mu$SR data. (b) Temperature dependent
comparison of the $\mu$SR local mean field, bulk magnetization $M$ measured by
SQUID magnetometry, and XMCD switching statistics for ZFC (green markers) and
FH-FC (red markers) statNote . The pictographs illustrate a muon stopping in
FiM clusters embedded in a PM sea. As the clusters grow in size at lower
temperatures, the average internal $B$-field shows a negative shift indicated
by arrow in (a).

## Methods

###  I Sample preparation

We have grown 200 nm Ge0.87Mn0.13Te thin films on InP(111) and BaF2(111)
substrates by molecular beam epitaxy. The choice for 13% Mn-doping is that for
such a doping the Zeeman splitting is saturating JK_GMT . A protective stack
of amorphous Te- and Se-capping layers with a total thickness of $\approx$20
nm was used to avoid surface oxidation and degradation. Before mounting into
the XTREME beamline sample holder Piamonteze_XTREME_2012 , it was completely
removed in a ultrahigh vacuum chamber by annealing the samples for 30-45 min
at 250$\,{}^{\circ}\mathrm{C}$; and recapped with $\approx$2 nm amorphous
carbon by carbon-thread evaporation within the same vacuum chamber.

###  II Experimental methods

The XMCD setup at the XTREME beamline Piamonteze_XTREME_2012  allows us to
measure magnetic moments from the averaged orbital and spin contribution
projected along the out-of-plane magnetization $\vec{M}$, which is collinear
with the ferroelectric polarization $\vec{P}$ along the [111] direction JK_GMT
. Raw TEY and TFY signals measured with Keithley 6517A digital multimeters
were normalized with the $\mathrm{I_{0}}$ photocurrent from the XTREME
beamline refocussing mirror. All the signals were sampled from digital
multimeter analog outputs, and further processed for the on-the-fly XMCD
energy scans with a dedicated VME-system readout system by sampling/averaging
the signals with 1 kHz and 100 Hz, respectively JK_OTF_AIP_2010 . The XMCD
spectrum was obtained by taking the difference of ($\mu$\+ - $\mu$-, where
$\mu$\+ and $\mu$\- represent the XAS spectra measured by TEY or TFY and using
the circular polarized X-rays with variable magnetic field up to 6 T parallel
and anti-parallel to the beam direction. All data presented were measured
during three independent beamtimes on fresh samples prepared in exactly the
same way. The same samples were used for further NEXAFS (SuperXAS beamline at
Swiss Light Source) and SQUID magnetometry. Special extra-large samples capped
with Au were prepared for low energy muon spin relaxation experiment
(LE-$\mu$SR) at the $\mu$E4 muon beamline at the Paul Scherrer Institut
Prokscha2008NIPA . LE-$\mu$SR time spectra are measured with two positron
detectors placed opposite to each other for the considered samples.
Temperature dependent asymmetry spectra were obtained under an external
magnetic field of 5 mT applied perpendicular to the initial muon spin
polarization direction. Data indicate magnetically ordered islands with a net
magnetic moment aligned with the applied field, embedded in a sea of
paramagnetic environment. As these islands grow in size, their magnetic moment
along the applied field increases and the strength of their dipolar field in
the paramagnetic regions increases. This dipolar field is primarily opposite
the direction of the applied field and hence produces the observed negative
shift in B(T) as seen in Fig. 6a.

In order to achieve comparable local probe conditions with the XMCD
photoelectron escape depth ($\approx$20 nm), the muon implantation energy was
adjusted to set their stopping distribution just below the sample surface.
Under magnetization, the asymmetry of the transverse field the muon precession
decays. The main quantity determining the LE-$\mu$SR spectra in Fig. 6a is the
muon spin precession amplitude decay from asymmetric emission of positrons,
analyzed with Musrfit program Musrfit_2012 .

###  III Crystal structure and computational details

Using first-principles calculations within the density functional theory, we
have investigated the ground state magnetic properties of bulk
Ge0.867Mn0.133Te. The calculations were carried out using the multiple
scattering KKR Green function method as implemented in the spin-polarized
fully relativistic Korringa-Kohn-Rostoker (SPRKKR) code Minar_RPP . The
exchange and correlation effects were incorporated within the LDA framework
Vosko_1980 . Brillouin zone integrations were performed on a
39$\times$39$\times$39 dense mesh of $k$-points. The angular momentum
expansion up to lmax= 4 has been used for each atom. The energy convergence
criterion and coherent potential approximation (CPA) tolerance has been set to
10-7 Ry. The potential is constructed in full potential geometry. The
influence of chemical disorder in Ge0.867Mn0.133Te can be estimated by means
of the CPA alloy theory Soven67 ; Taylor67  implemented in the SPR-KKR method.

We consider two types of $\mathrm{Mn_{i}}$ atoms denoted as $\mathrm{Mn_{2}}$
and $\mathrm{Mn_{3}}$ in Fig. 2a, which occupy allowed Wyckoff positions
within the Ge0.87Mn0.13Te space group (see Table 1), derived from the
experimental data as determined from the room temperature X-ray powder
diffraction measurements Johnston_1961 .

Ge0.867Mn0.133Te
---
space group: R3mR (No. 160), $a$=4.234Å, $\alpha$=59.07∘

Wyckoff sites with atomic positions |  Atomic occupancies
---|---
1a (0.534,0.534,0.534) | 0.911Ge + 0.089$\mathrm{Mn_{s}}$
1a (0.0,0.0,0.0) | Te
1a (0.250, 0.250, 0.250) | 0.978Es + 0.022Mn2
1a (0.764, 0.764, 0.764) | 0.978Es + 0.022Mn3

Table 1: Ge0.867Mn0.133Te R3mR (No. 160) space group crystal structure with atomic positions and occupancies. $Es$ represents empty spheres used in the CPA formalism, $\mathrm{Mn_{i}}$ with two atomic positions (Mn2,Mn3) occupying allowed Wyckoff positions of R3mR space group. The atomic distribution of Mn atoms as substitutional (Mn2; 67%) and interstitial (Mn2,Mn3; 33%) were obtained from NEXAFS measurements summarised in Extended Fig. 1c. CPA theory $\mu_{B}$/atom |  XMCD ($\mathrm{Mn_{s}}$+$\mathrm{Mn_{i}}$) $\mu_{B}$/atom
---|---
| $\mathrm{Mn_{s}}$ | $\mathrm{Mn_{i}}$ | $\Sigma\mathrm{Mn_{s,i}}$ | MSR-off 0 T | MSR-on 0 T | MSR-on 6 T
$\mathrm{m_{S}}$ | 4.209 | -2.112 | 2.097 | 0.058$\pm$0.010 |  $\approx$-0.089 |  $\approx$0.238
$\mathrm{m_{L}}$ | 0.047 | -0.079 | -0.032 | 0.040$\pm$0.004 |  $\approx$-0.014 |  $\approx$-0.012
Table 2: Spin ($\mathrm{m_{S}}$) and orbital ($\mathrm{m_{L}}$) magnetic
moments CPA theory vs. XMCD from Ge0.87Mn0.13Te/InP[111], $\mathrm{m_{S}}$ and
$\mathrm{m_{L}}$ momenta were calculated along the Ge0.87Mn0.13Te[111] easy
magnetization axis.

In addition we also performed 2$\times$2$\times$2 super-cell QuantumESPRESSO
pseudopotential relaxation calculations with the Mn atomic positions
summarized in Table 1. The results confirmed that AFM is the energetically
most stable configuration. The structural models from the super-cell
calculations were used to model XAS and XMCD calculations, which in turn where
consistent with our CPA model summarized in Fig. 2d. Moreover, the structural
relaxations around the Mn atoms showed negligible impact on the magnetic
structure and shape of the XAS spectra.

Finally, we also performed multiplet calculations based on localized Mn2+ $3d$
with parameters reported by Sato et al. Sato_JPRPh_2005 . The comparison with
measured Ge0.87Mn0.13Te XAS spectra in Extended Fig. 1c and Fig. 2a confirms
that our XAS and XMCD spectra feature localized Mn2+ $3d$ states. Therefore,
for the XMCD sum rule calculations we considered 5 holes in the Mn unoccupied
$d$-states with a correction factor of 1.47 for compensating the $jj$-mixing
Goering_2005 . However, as reported by Sato et al. Sato_JPRPh_2005 , multiplet
calculations suffer from intensity deviations between the calculations and
experiments. For a better comparison with experimental data we thus relied on
CPA calculations.

Because the CPA enables us to address spectral contributions from individual
Mn-atoms, we tentatively modelled the resulting magnetic dichroism as a simple
sum of all Mn contributions in saturation, as shown in Extended Fig. 2b. When
reducing the $\mathrm{Mn_{s}}$ spectral contribution a factor 3, we find that
the resulting XMCD signal (dashed line in panel d) reproduce the measured data
very well, whereas multiplet calculations with $\mathrm{Mn_{s}^{2+}}$ atoms
fail to reproduce the exact shape of the dichroism effect and lead to a
spectral shift indicated by the black arrow (see Extended Fig. 2d). The factor
3 reduction of the $\mathrm{Mn_{s}}$ spectral contribution is rationalized by
observing a significant decrease in $\mathrm{Mn_{s}}$ dichroism between state
\small{1}⃝ right after the FH-FC@6T at 10 K, and \small{2}⃝ recorded later
during MSR-on conditions (vertical arrow in Extended Fig. 2b). With every FiM
switching event at low temperatures some of the $\mathrm{Mn_{s}}$ becomes part
of the paramagnetic spin glass background; and when around one third of the
original dichroism effect is left, this process appears to saturate (Extended
Fig. 2c). However, on increasing the temperature from 10$\rightarrow$45-65 K,
we observe a significant increase in the $\mathrm{Mn_{s}}$ dichroism, which we
attribute to self-induced magnetization stimulated by MSR along with the
paramagnetic spin glass background as schematically depicted in Extended Fig.
4b.

###  IV Magnetic ground state calculations

Figure 2a is a cartoon descriptions of various Mn-Mn exchange interactions
between individual $\vec{S_{1}}$, $\vec{S_{2}}$ spin states considered in our
model Hamiltonian where Mn impurities act at the same time as random magnetic
moments and as acceptors producing the carriers. For the Ge0.87Mn0.13Te
magnetic ground state we consider the general spin Hamiltonian in the
following form:

$H=-\frac{1}{2}\sum_{ij}J_{,ij}\vec{S_{i}}\cdot\vec{S_{j}}+\sum_{ij}\vec{D_{i,j}}\cdot(\vec{S_{i}}\times\vec{S_{j}})+\sum_{i}K_{i}\vec{S_{i}}^{2}$
(1)

The first term represents the Heisenberg exchange energy with $\vec{S_{i}}$
and $\vec{S_{j}}$ unit vectors having directions corresponding to local
magnetic moments on sites $i$ and $j$. $J_{i,j}>0$ and $J_{i,j}<0$ prefer FM
and AFM spin configurations, respectively. The second term originates from the
antisymmetric part of the interaction matrix, also termed as antisymmetric
exchange or Dzyaloshinskii-Moriya interaction (DMI). Finally the $K_{i}$ term
in the final term accounts for the magnetocrystalline anisotropy energy (MAE).

The isotropic $J_{i,j}$ exchange between localized $\vec{S_{1}}$,
$\vec{S_{2}}$ spin states depends on the distance between the next-nearest Mn
dopants. In contrast to earlier theoretical Ge1-xMnxTe studies considering
only $\mathrm{Mn_{s}}$ atoms Fukushima_2014 , another type of interaction is
coming from the $\sum_{i,j}\vec{D_{i,j}}\cdot(\vec{S_{i}}\times\vec{S_{j}})$
Dzyaloshinskii-Moriya interaction obtained by sampling the magnetization
deflection between [111] and [$\overline{111}$] easy magnetization axis
directions. Below we discuss the contributions of individual terms to the
Ge0.87Mn0.13Te magnetic ground state.

#### Isotropic Exchange Interactions

The magnetic exchange coupling parameters ($J_{i,j}$) are based on the real
space approach by using the theory proposed by Liechtenstein et
al.Liechtenstein87 . This approach employs the “magnetic force theorem” to
determine $J_{i,j}$ by assessing the total energy change related to an
infinitesimal rotation of the magnetic moments located at the atomic sites $i$
and $j$. The energy change can be related to the exchange coupling parameters
$J_{i,j}$ as:

$J_{i,j}=\frac{1}{4\pi}\int^{E_{F}}dEImTr_{L}{\\{}{\Delta_{i}\tau^{i,j}_{\uparrow}\Delta_{j}\tau^{j,i}_{\downarrow}}{\\}}$
(2)

where $\tau$ is the scattering path operator, $\Delta_{i}$ is the difference
in the inverse single site scattering $\it{t}$ matrices for spin up and spin
down electrons, and TrL is the trace of scattering matrices over the orbital
indices $\it{L={(l,m)}}$.

The $J_{i,j}$ calculations are performed within a cluster of radius 3$a$,
where $a$ is the lattice parameter. We neglect all interactions involving Ge
and Te atoms and consider only those between Mn atoms which are found to host
significant localized magnetic moments/atom. Fig. 2a of the main text shows
three different types of Mn atoms inside the Ge0.867Mn0.133Te primitive unit
cell, classified as substitutional $\mathrm{Mn_{s}}$ and interstitial
$\mathrm{Mn_{i}}$ atoms, respectively.

The physical mechanism behind the magnetic ground state has been attributed to
$pd$-exchange coupling Fukushima_2014 ; Belhadji_2007 , combined with $d$-$d$
magnetic interactions Sato_JPRPh_2005 . Consistently with our theoretical
predictions, the $pd$-exchange coupling is relatively weak, but long ranged.
Moreover, for Ge1-xMnxTe the $pd$-exchange constant is negative
Sato_JPRPh_2005 , stemming from AFM exchange interaction between
$\mathrm{Mn^{2+}}$ states. In agreement with our Ge1-xMnxTe magnetic ground
state description, this negative term naturally materialize between
$\mathrm{Mn_{i}}$-$\mathrm{Mn_{s}}$ which is further confirming that
Ge1-xMnxTe is a ferrimagnetic rather than a ferromagnetic DMS Fukuma_2005 ;
Sato_JPRPh_2005 ; Fukuma_2006 ; Fukuma_2008 ; Fukushima_2014 .

#### Dzyaloshinskii-Moriya interaction

The DMI is a chiral exchange interaction between localized spins and has a net
contribution only in systems without structural inversion symmetry and with
the presence of an indirect or long-range exchange interaction. Ge0.87Mn0.13Te
satisfies these two conditions, which is also evident from the isotropic
exchange interactions described in the previous section. The origin of DMI is
found in the spin-orbit coupling (SOC) which acts as a perturbation on
localized orbital states. Given two neighboring spins $\vec{S_{1}}$ and
$\vec{S_{2}}$, the DMI energy can be described by
-$\vec{D_{1,2}}\cdot(\vec{S_{1}}\times\vec{S_{2}})$. Therefore, the direction
of the relative $\vec{S_{1}}$ and $\vec{S_{2}}$ rotations can be clockwise or
counter clockwise, providing the information about helicity in the case of
spin spirals. This expression is part of a generalized exchange interaction
where the DMI term is related to the exchange constant $J$ of the direct
Heisenberg exchange $-J(\vec{S_{i}}\cdot\vec{S_{j}})$. However, contrary to
the latter which favors collinear alignment, the DMI promotes an orthogonal
arrangement between $\vec{S_{i}}$ and $\vec{S_{j}}$, with a chirality imposed
by the direction of $\vec{D_{i,j}}$. The resulting spin helicity is uniform,
meaning that clockwise or counterclockwise rotation of spins are energetically
identical. Data in Fig. 2c show all the components of $\vec{D_{i,j}}$ (i.e.
$D_{i,j,x}$,$D_{i,j,y}$,$D_{i,j,z}$) interaction, showing robust exchange up
to a cluster size of 3.0$a$. The average DMI energy is comparable to the
$J_{i,j}$ exchange energy for larger distances; and a factor 4 higher than the
magnetic anisotropy energy (MAE) discussed below.

#### Magnetic Anisotropy Energy

The magnetic ground state of FiM exchange energy (grey markers in Fig. 3a) has
been explored by determining the total energy corresponding to the relative
spin orientation of $\mathrm{Mn_{2}}$ and $\mathrm{Mn_{3}}$ spins with respect
to the substitutional $\mathrm{Mn_{1}}$ spins. The $\mathrm{Mn_{1}}$ spin was
frozen along the [111] quantization axis and the $\mathrm{Mn_{2}}$,
$\mathrm{Mn_{3}}$ spins were rotated along the out-of-plane [111] direction
starting from $\theta$=0, which corresponds to the FM state with all spins
pointing in the same direction. However, the self consistency convergence in
calculations can be achieved only for a limited $\theta$ range between
40∘-320∘. Nevertheless, the total energy of the system reaches a minimum with
$\mathrm{Mn_{1}}$ spins set to 0∘ and $\mathrm{Mn_{2,3}}$ spins to 180∘, which
establishes the FiM order. The red markers in Fig. 3a show the MAE energy
variation as a function of magnetization angle $\theta$, evaluated as a
difference between the fully relativistic total energies calculated for
quantization axes [111] and axis orthogonal to $\bar{[111]}$. Data confirm
that the easy magnetization axis is along the [111] or
[$\bar{1}\bar{1}\bar{1}$] directions. The calculated MAE of the
Ge0.867Mn0.133Te is 0.101 meV/f.u. and represents the energy barrier between
the easy and hard magnetic axes. Finally, the energy landscape obtained by
combination of MAE and FiM ground state was used to model the FiM double-well
potential as depicted in Fig. 3a. The MSR-driven FiM switching contracts the
states into one of the two wells. The bistable potential depicted can be
expressed as $U(x)=-\frac{1}{2}ax^{2}+\frac{1}{4}bx^{4}$ with minima located
at $\pm$90∘ and barrier height $\Delta V=a^{2}/4b$.

## Extended data

Figure 1: (a) Stacked plot of $\mathrm{Mn_{L3}}$ XMCD spectra measured with
same circular light polarization $\mu^{+}$ under MSR-on regime. A change in
x-ray helicity relative to the fixed magnetization direction is equivalent to
a change in the magnetization direction relative to the fixed helicity. (b)
Spontaneous switching in consecutive XMCD datasets recorded in total
fluorescence yield (TFY) at T=5 K and B=0 T, zoomed at the $\mathrm{Mn_{L3}}$
and $\mathrm{Mn_{L2}}$ absorption edges. (c) NEXAFS Mn-K edge data fits. Panel
inset shows the best fit obtained with $\mathrm{Mn_{s}}$:$\mathrm{Mn_{i}}$
occupancy close to 2:1. Figure 2: (a) Background subtracted XAS spectra
measured at 6 T (10 K), compared with multiplet calculations. (b)
$\mathrm{Mn_{L3}}$ XMCD measured after ZFC at 1 T (blue trace), compared with
FH-FC spectra in $\mathrm{Mn_{s}}$ state \small{1}⃝ and \small{2}⃝,
respectively. (c) Time dependence of the dichroism effects under MSR-on regime
at $\mathrm{Mn_{L3}}$-edge between the initial magnetic state \small{1}⃝ after
FH-FC and magnetic state \small{2}⃝. (d) Calculated XMCD spectra using CPA and
multiplet calculations. For CPA spectral contributions from individual Mn
atoms are indicated. CPA is in good agreement with the experimental data,
whereas the multiplet calculations show a clear shift in the spectral features
as indicated by the horizontal arrow. Figure 3: (a) TEY time survey of
$\mathrm{Mn_{L3}}$ pre-edge (red markers) and edge (blue markers) obtained
during XMCD acquisition. (b) TEY time survey at $h\nu=638.8$eV
($\mathrm{Mn_{L3}}$ edge) overlaid with the machine top-up current. (c) Series
of TEY time surveys at $h\nu=638.8$eV under applied B-field and different
x-ray polarizations. Figure 4: (a) (top) Magnetostochastic switching residence
time distribution $N(t)$ for a data set measured at 5 K, ZFC and $t_{\Omega}$
periodic drive obtained by measuring full XMCD spectra covering both
$\mathrm{Mn_{L3}}$+$\mathrm{Mn_{L2}}$; compared with all data measured after
FH-FC for 5, 45 and 65 K (red markers in Fig. 6b) where the switching
statistics was evaluated from shorter energy scans across the
$\mathrm{Mn_{L3}}$ only, i.e. by halving the periodic drive
($\mathrm{\frac{1}{2}t_{\Omega}}$). The arrows highlight the $N(t)$ phase
shift due to different stochastic resonance conditions Gammaitoni_RMP_1998 .
(b) Series of $\mathrm{Mn_{L3}}$ XMCD spectra measured at 65 K, with maximum
dichroism effect close to 60%. The pictographs illustrate enhancement of the
FiM order (red-blue arrows) spreading across the magnetic clusters along with
self-induced magnetization of the PM sea under MSR FiM switching.