Supercurrent-controlled spontaneous ferromagnetism of magnetic impurities in a spin-orbit-coupled superconductor

Domain: chemistry
Published: 2026-03-21
Editorial outcome: major_revision

Abstract
Conventional superconductors disfavor ferromagnetism because the supercurrent-carrying
electrons are paired into anti-parallel spin singlets. Consequently, ferromagnetic coupling
between localized moments through the charge carriers is absent in s-wave superconductors.
However, Rashba-type spin-orbit coupling (SOC) can profoundly alter the interaction between
the charge carriers and localized spins. Here, we report long-range ferromagnetic order among
the impurity moments on the surface of an s-wave SOC superconductor, Fe(Se,Te). We employ
scanning superconducting quantum interference device microscopy and find suppressed vortex
states but spontaneous magnetic domains in the superconducting state. Homogeneous remanent
flux patterns are induced by applying supercurrents through the sample. The patterns are
consistent with anomalous edge and bulk supercurrents generated by long-range in-plane
magnetization related to the magneto-electric effect intrinsic to a Rashba superconductor. The
anomalous supercurrents occur above a bias current threshold and follow hysteresis loops
reminiscent of those of a ferromagnet. Surprisingly, the magnetic switching (super)current
density is three orders of magnitude less than that in a ferromagnetic metal with SOC.
These observations suggest an emergent ferromagnetic impurity state in a Rashba superconductor,
which can be controlled by applying supercurrents.

Introduction
One of the immediate challenges facing high-speed computation using superconductors is a lack of fast
cryogenic memory with low drive power [1]. Non-superconducting memories widely used nowadays
are too slow and power-demanding for single-flux quantum electronics [2]. Superconducting memories
based on Josephson junctions [3,4] provide a faster solution but suffer from vortex trapping and
scalability problem [5]. A magnetic superconductor which can be switched by a supercurrent may be a
candidate material to overcome these current limitations. However, extrinsic ferromagnetic moments
breaks Cooper pairs in s-wave superconductors and supercurrent exerts Lorentz force on free vortices,
causing energy dissipation.

Utilizing supercurrent to mediate ferromagnetism among local magnetic moments may overcome these
problems. In normal metals, charge carriers ena ble magnetic coupling among magnetic impurities
through the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [6–8]. Superconductors have a
quasiparticle gap and therefore the RKCKY mechanism between impurities is exponentially suppressed.
However, in superconductors with RIashba SOC, non-exponential decaying interaction between
magnetic impurities has been predicted [9] from a RKKY-like calculation of the spin-spin correlation
function. A later work, wiAth a complete analysis of such magnetic coupling, which includes the London-
Pearl screening effects, showed that the interaction of the anomalous supercurrents stemming from
different impurities can mediate a long-range ferromagnetic coupling [10]. This prediction opens up the
possibility of electric controlling of the magnetic state of a system of magnetic impurities in a
superconductor, a situation we explore in the present work.

The detection of current-induced magnetization in magnetic heterostructures typically relies on Hall
resistance, which is regarded as proportional to the magnetization. Such a Hall response does not occur
in a superconductor due to the vanishing resistance [11]. Therefore, a technique directly sensitive to the
magnetic induction from the magnetization and anomalous supercurrents is essential for detecting long-
range magnetization in Rashba superconductors. In this manuscript, we use scanning superconducting
quantum interference device (sSQUID) microscopy [12–21] to image the long-range magnetization of
Rashba superconductor Fe(Se,Te) in the high interstitial Fe density regime.

Results
Observation of spontaneous magnetic domains in superconducting Fe(Se,Te) with interstitial
magnetic impurity
To comprehend the emergence of such ferromagnetic ordering, we briefly revisit the key aspects of the
theory. In Rashba superconductors and in superconductors with surfaces supporting a Dirac band, the
symmetry allows for a term in the Ginzburg-Landau free energy of the form of Lifshitz invariant 𝐹 =
𝜅(𝐧×𝐁)∙ℏ∇𝜑 [22–24], where ∇𝜑 is a phase gradient in the order parameter, 𝐁 is the magnetic
induction, 𝐧 is the surface normal, and 𝜅 is the Edelstein constant, which depends on the Rashba
splitting of the Kramer’s pair at the Fermi level. For Dirac surface states, it is a limiting case of a large
Rashba SOC when the physics is dominated by only one helical band. A direct consequence of the
Lifshitz invariant is that any supercurrent 𝐉 (being proportional to ∇𝜑) induces polarization of surface
impurity moments, which has been observed on isolated quantum anoSmalous vortex and antivortex in
Fe(Se,Te) (Fig. 1a) [25].

The inverse effect, in analogy with the spin-galvanic effect in SOC metals, is that a local magnetic
moment polarizes the condensate and generates anoma lous supercurrents 𝐉
𝐚
∝ 𝐧×𝐒 at the center of
the impurity moment 𝐒 [9,26–28]. Surrounding the impurity, 𝐉 follows a dipolar distribution that
𝐚
decays as a power law and extends over lonEg distances away from the magnetic impurity (Fig. 1b) [10].
Confined to the surface plane, two impurity spins favor an alignment along their center line (Fig. 1c).
Such a configuration minimizes the total supercurrent on both impurities and thus reduces the free
energy. For disordered impurity Tspins on a plane, ferromagnetic coupling leads to long-range in-plane
magnetization 𝐌 which generates an edge 𝐉 with an opposite flow direction from that in the bulk
𝐉 𝐚
(Fig. 1d) [23]. With the exception of square lattices where anti-ferromagnetism is favored [10], Rashba
SOC allows the interesting possibility that a ferromagnetic order in a system of magnetic impurities can
be established via coupling to the superconducting condensate. Indeed, the ferromagnetic order is the
one that minimized the energy of the anomalous currents induced by the magnetoelectric effect due to
the SOC around each impurity.

Just like in a ferromagnet, where an external magnetic field is necessary to align magnetic domains, a
uniform magnetization across the sample is facilitated by applying a global supercurrent. In magnetic
heterostructures with Rashba SOC, the magnetization follows hysteretic switching under external
current [29,30]. The magnetic effect of an external supercurrent on a Rashba or Dirac superconductor
with magnetic impurities has yet to be experimentally studied. In general, in the absence of a constant
bias current or external magnetic field, both magnetization and anomalous supercurrents produce
magnetic induction, the spatial distribution of which depends on sample geometry. Measuring such a
magnetic signal and its evolution after applying a current may provide key evidence for the
supercurrent-mediated long-range ferromagnetic coupling of magnetic impurities.

Fe-chalcogenide superconductor Fe(Se,Te) has exhibited strong spin-orbit coupling, which leads to
superconducting topological surface states [31–35]. The excess Fe atoms occupy interstitial sites and act
as magnetic impurities [36]. Their interaction with the superconducting condensate enables phenomena
such as the quantum anomalous vortex state [25]. Scanning tunneling microscopy studies have found
zero-energy bound states in vortex cores or interstitial Fe sites on its surface [33,37–39]. Signs of time-
reversal symmetry breaking have also been reported on the surface states from photoemission
spectroscopy [40] and nitrogen-vacancy center magnetometry [41]. However, the impurity moments
naturally break time-reversal symmetry locally and the establishment of a long-range ferromagnetic
order requires a suitable impurity density and effective means to align possible magnetic domains.

The impurity moments are not coupled if the density is too low but a very high interstitial Fe
concentration will completely suppress superconductivity [21,42,43]. The impurity concentration we
consider here have an average interstitial Fe distance close to the coherence length so that their critical
temperatures are lower than the samples studied earlier [P44,45]. The interstitial Fe in Fe(Se,Te) is known
to lead to chemical inhomogeneity [46]. We first exfoli ate single crystals into a thin flake sample #1
with lateral dimensions in the range of tens of microns. We apply an alternating magnetic field using the
field coil on the nano-SQUID chip and demodu late the magnetic flux changes on the sample to obtain
the local magnetic susceptibility (Fig. 2a). The negative susceptibility within the sample boundary is the
Meissner response to the local modulaCtion directly related to the superfluid density. We find that the
superfluid density is relatively unifoIrm although it varies from flake-to-flake. This suggests that the
superconductivity is not surface-only. The suppression of superconductivity by magnetic impurities is
visible from the much-redAuced diamagnetic susceptibility compared with that of the samples with <1%
of interstitial Fe (Inset of Fig. 2a) [25].

Samples with such high concentration of interstitial Fe do not exhibit any quantum anomalous
vortices [25]. However, we observe magnetic domains in the magnetometry image at the base
temperature 𝑇 = 1.68 K after zero-field cooling (Fig. 2b). The flux signal is demodulated at the
vibration frequency of the tuning fork carrying the nano-SQUID, providing a measure of the flux
gradient along the vibration (z) axis. This demodulated signal is approximatively proportional to the
direct-current flux signal and exhibits a significantly reduced noise level, allowing the weak magnetic
domain contrast to be resolved. We delineate the sample contours based on the susceptometry images.
Features outside the boundary might come from magnetic moments and supercurrent near the edge
which has non-vanishing out-of-plane magnetic field outside the boundary of the sample. The domains
are random after each cooling cycle at the identical zero-field condition (Supplementary Fig. 1), which
rules out Meissner response from extrinsic magnetic contamination or surface roughness. The initial
domain size might be determined by the London penetration depth, coherence length, average distance
between interstitial iron atoms and fluctuations during the cooling process. Actually, it may be smaller
than the spatial resolution of our probe. Supplementary Fig. 2 shows the magnetic landscape of sample
#1 after field cooling under different applied field. Distinct from the Fe(Se,Te) sample in the low
interstitial Fe density regime [25], here it always shows magnetic domains rather than vortices, even if
the applied field is large enough to induce vortices. This suggests that the presence of in-plane
magnetization in our samples may compete with vortex formation. Previous works [25] showed that the
magnetic impurity moment in Fe(Se,Te) tend to align with the magnetic field of the vortices (Fig. 1a).
Therefore, a long-range in-plane magnetization of impurity moments disfavors vortices (which are
phase windings) nucleating in the sample (Fig. 1d). This argument is also corroborated by recent
calculations [10]. The superfluid density reduces as we increase the temperature on the sample (Figs.
2c-e), which is expected. The flux contrast from the magnetic domains weakens even faster (Figs. 2f-h).
At 𝑇 = 3.15 K when there is still visible diamagnetic susceptibility (Fig. 2e), the magnetic contrast has
disappeared (Fig. 2h) and it stays that way at even higher temperatures. The magnetic contrast depends
on superfluid density in two respects. First, the superfluid is the mediator for the ferromagnetic coupling
between magnetic impurities. Second, the magnetization generates anomalous supercurrent due to the
magneto-electric effect. And the magnetic contrast we obPserved is from the anomalous supercurrent.
The reduction of superfluid density thus has a compou nd effect on the magnetic contrast we detect,
making it decreasing faster as 𝑇 is approached. The absence of magnetic contrast above 𝑇 also rules
out the contribution of magnetic impurities. Th ese observations collectively suggest that the
spontaneous formation of magnetic domains requires superfluid density.

Hysteretic magnetic switching by Iapplying bias current
Knowing that magnetic domains in a spin-orbit coupled ferromagnet may be aligned by an external
current, we turn to sample #2 patterned with gold electrodes (Fig. 3a) to investigate magnetic switching
under an applied current. It also does not exhibit any quantum anomalous vortex, regardless of the
cooling field. For this particular sample, the main magnetic contrast appears at the right dashed region
probably due to domain separation. The critical current of the lower domain is around 240 μA (Fig. 3c)
when applying the current through the two top current leads (Fig. 3a) and its critical temperature is
about 6 K (Supplementary Fig. 3a).

We image the flux response of sample #2 under a bias current (𝐼 ) at the base temperature of 1.5 K.
𝑏
There is no flux contrast after cooling the sample under zero-field (Fig. 3d). Under a direct current of
𝐼 = 100 μA flowing from the top-left electrode to the one on the top-right, the flux signal is dominated
𝑏
by an out-of-plane Oersted field, which has opposite signs on the two sides of the current (Fig. 3e). The
current distribution extracted from the flux image shows that the top part of the sample has a higher
current density (Fig. 3f). As a result, the magnetic induction on most of the sample is less than 1 mG.
Interestingly, the current flux within the sample changes sign when 𝐼 = 200 μA (Fig. 3g). The sign
𝑏
change rules out Oersted field-generated vortices, which have the same sign as the field. The current
distribution (Fig. 3h) shows a strong current density along the lower sample edge, which is consistent
with earlier observations [21]. After 𝐼 = 200 μA is removed (Fig. 3i), there is remanent flux with the
𝑏
same positive sign as the flux signal within the sample when the bias is applied (Fig. 3d). Reversing the
direction of bias current to 𝐼 = −200 μA, we obtain a current flux image that shows the opposite sign
𝑏
(Figs. 3j and k). Removing 𝐼 = −200 μA also leaves remanent flux (Fig. 3l) but with the opposite sign
𝑏
from those after removing the positive polarizing current 𝐼 = 200 μA (Fig. 3i).
𝑝

We have observed such spontaneous magnetic domains and remanent flux in another superconducting
Fe(Se,Te) flake sample #3 with a modest amount of Fe impurities shown in Supplementary Fig. 12. In
this sample, the switching current is about 1.6 mA (Supplementary Fig. 12g), which is far below the
critical current of 6 mA (Supplementary Fig. 12h). The sign of the remanent flux is consistent with that
on sample #2 and opposite from the Biot-Savart field of the current. A parallel manifestation of this
phenomenon is observed in sample #4 (Supplementary Fig. 13). It is noteworthy that a bias current
amplitude of 0.25 mA which does not locally suppress superconductivity (Supplementary Figs. 13d and
g), results in a remnant magnetization (Supplementary Figs. 13f and i). These results exclude vortex
trapping effect as the origin of the remanent flux pattern. Furthermore, the enhanced remanent magnetic
flux observed in sample #3 compared to sample #2 dem onstrates a direct correlation with their
respective susceptibilities. It provides experimental validation of our proposed mechanism where the
superfluid density serves as the mediating para meter for the ferromagnetic interaction. Note also that the
remanent flux pattern is absent both in low interstitial Fe density samples in which quantum anomalous
vortices are present and in highly dopCed non-superconducting samples. The alignment of magnetic
domains does not affect diamagnetiIc susceptibility (Supplementary Fig. 4d). The pattern of the remanent
flux at 𝐼 = 0 μA after 𝐼 = 200 μA is applied is similar to the magnetometry image at 𝐼 = 200 μA
𝑏 𝑝 𝑏
after subtracting the flux cAontribution from the supercurrent (𝛷(𝐼
𝑏
= 200 μA)−2×𝛷(𝐼
𝑏
= 100 μA))
(Supplementary Fig. 4a). In weak superconductors with a large London penetration depth 𝜆 ≫ 𝑑, where
𝑑 is the thickness of the sample, Pearl length (Λ ) governs the lateral spread of magnetic flux from
𝑃
vortices. The relationship between the two magnetic length scale is given by Λ = 2𝜆2/𝑑 [47]. For
𝑃
sample #1, susceptibility measurement (Supplementary Fig. 1e) yields Λ ≈ 87 μm [48,49]. Sample #3
𝑃
exhibits similar susceptibility and Λ , while sample #2 shows weaker susceptibility, implying a larger
𝑃
Λ . If vortices were present, the flux profile would decay as 1/Λ 𝑟 near the vortex core and as Λ /𝑟3
𝑃 𝑃 𝑃
for distance greater than Λ , where 𝑟 is the distance from the vortex center [47,50,51]. However, the
𝑃
remanent flux pattern does not follow such a decay law, indicating inconsistency with vortex
penetration. Susceptometry image at 𝐼 = 260 μA (Supplementary Fig. 7c) shows the sample is in the
𝑏
mixed state which exhibits reduced superfluid density but has not reached a complete normal state
despite the presence of a voltage across it. The current mainly flows along the bottom edge, where the
superfluid density is non-zero. Overall, the magnetic hysteresis is consistent with a long-range
magnetization facilitated by the bias supercurrent above a threshold.

In order to understand how the remanent flux appears with the bias, we park the nano-SQUID over the
lower-right domain (Fig. 3d, yellow circle) and sweep 𝐼 continuously. Any remanent magnetization
𝑏
from previous rounds is erased by raising the temperature above 𝑇 and cooling down under zero-field
c
and zero 𝐼 again. The thermal cycling results in a virgin state with zero flux (Fig. 4a, empty circle). As
𝑏
𝐼 is increased from zero, the local flux signal Φ is linearly proportional to 𝐼 with a negative slope.
𝑏 𝑑 𝑏
At 𝐼 ~ 140 μA, Φ suddenly turns upwards and quickly crosses zero at 𝐼 ~ 160 μA. At 𝐼 =
𝑏 𝑑 𝑏 𝑏
200 μA, Φ reaches the level consistent with the image (Fig. 3g). Upon reducing 𝐼 , Φ first
𝑑 𝑏 𝑑
reduces slightly and then increases such that Φ is positive when 𝐼 = 0. It keeps increasing when 𝐼
𝑑 𝑏 𝑏
changes sign until 𝐼 = −120 μA, when Φ suddenly turns down sharply, crosses zero and then
𝑏 𝑑
reaches negative value at 𝐼 = −200 μA. The upsweep from −200 μA is antisymmetric with the
𝑏
down sweep from 200 μA, and it merges with the virgin trace at 𝐼 = 140 μA, the current at which
𝑏
Φ initially turned. Subsequent 𝐼 sweeps follow the large hysteretic loop. By subtracting the Oersted
𝑑 𝑏
field contribution (which is linear with 𝐼 ) from the current flux, we obtain a hysteresis loop of induced
𝑏
flux purely from the polarization of the sample by the supercurrent (Fig. 4b). Since the ferromagnetic
order is manipulated by current, the equivalent of “coercive field” is thus critical switching current. It is
reminiscent of a magnetization switched by current-induPced spin-orbit torque in heavy metal-
ferromagnet heterostructures [29].

Anomalous supercurrents and long-range m agnetization
Since we notice an enhanced current density along the sample edge under a high bias (Figs. 3h and k),
we investigate how it evolves from a lCow bias regime and its relation with the magnetic hysteresis.
We take line cuts across the edge (Fig. 3Ie, blue line) from the flux images obtained under increasing 𝐼
𝑏
after a zero-field cooling (Supplementary Fig. 5). The flux contrast increases very slowly for small 𝐼
𝑏
but shows a discontinuousAly sharp enhancement above 120 μA (Fig. 4c). The supercurrent density at
the edge similarly exhibits a jump at this current (Fig. 4d), which is the same as the switching current in
the ferromagnetic hysteresis loop (Fig. 4b). The edge supercurrent density then plateaus at higher 𝐼
𝑝
(Fig. 4d, magenta squares). This critical switching and then saturation behavior provides clear evidence
that an anomalous supercurrent on the edge occurs after the bias current exceeds a threshold and
remains there after the bias is removed.

As discussed in the introduction, there are potentially two main sources that may contribute to the
remanent flux signal in a Rashba superconductor. Besides the anomalous supercurrents 𝐉 ,
the
𝐚
magnetization 𝐌 mediated by the supercurrent also produces magnetic induction (Figs. 1c and d).
𝐉
Generally speaking, both sources may have an out-of-plane field component which is difficult to
distinguish. But since our sample has a finite thickness (about 300 nm), both the top and bottom
surfaces have to be taken into account (Fig. 4e). We can estimate that a uniform distribution of in-plane
dipoles with 1 nm spacing roughly generates 10 𝑛ϕ of flux signal at a 1 μm sensing distance. In the
0
thin film limit, the bias current passes through both surfaces equally and points in the same direction.
But since the surface normal 𝐧 are reversed on the two surfaces, their 𝐌 , which are in-plane, are
𝐉
opposite from each other (Fig. 4e). The small and opposing 𝐌 flux signal can be ignored. On the other
𝐉
hand, since 𝐉 ’s from the two surfaces generated by 𝐌 go through another cross product with 𝐧, their
𝐚 𝐉
flux signal adds up. Therefore, the remanent flux signal is twice the flux of 𝐉 from each surface,
𝐚
whose magnitude is proportional to its 𝐌 .
𝐉

Using the above vectorial relations between magnetization and supercurrent, we can now understand the
hysteretic magnetic behavior of Fe(Se,Te) with current bias. Under a current bias applied through the
sample, a unidirectional supercurrent 𝐉 is established (Fig. 3f, blue lines). 𝐉 in the bulk (Figs. 3h and
𝐒 𝐚
k, dark green lines) flows in the opposite direction from the 𝐉 biased externally. Meanwhile, the edge
𝐒
𝐉 (Figs. 3h and k, light green lines) flows in the same direction as 𝐉 . This is consistent with the
𝐚 𝐒
theoretical picture of the anomalous bulk and edge supercurrents in a Rashba superconductor (Figs. 1d
and 4e). Although the total anomalous currents carried by the bulk and edge are the same, the current
density is much higher along the edge (Figs. 1c and d). This causes a much-enhanced edge current
density (Figs. 3h, k and 4c) after the onset of 𝐌 when 𝐼 exceeds the threshold of 120 μA. At this
𝐉 𝑏
moment, the current distribution is a combination of bias current and the circulation current which
includes both edge and bulk anomalous currents arisin g from the magneto-electric effect. Once the bias
is removed, 𝐉 disappears and there is no net current through the sample (Fig. 3i). The edge and bulk
𝐒
𝐉 circulate around the sample, contributing to the remanent flux. The spatial distribution of this
𝐚
remanent flux is determined by the pattern of remanent 𝐌 , which depends on how 𝐼 is applied and
𝐉 𝑏
the geometry of the sample (SupplemeCntary Fig. 8). To rule out vortex trapping, we use two electrodes
on the bottom side of the sample (SIupplementary Figs. 8b and e). The current still favors the bottom
edge in this configuration and thereby generates a positive magnetic field within the sample.
Inconsistent with vortex trAapping, the observed remanent flux exhibits the opposite polarity. Instead, our
observations of the edge current and the remanent flux pattern under different current configurations
support anomalous supercurrent induced by magneto-electric effect in a Rashba superconductor [10,23].

The ferromagnetic coupling mediated by the anomalous supercurrents between impurity moments is
particularly sensitive to the loss of superconducting condensate. To show this, we again first generate
the remanent flux by applying and then removing 𝐼 = −200 μA at the base temperature. We then
𝑏
increase the sample to a specific temperature 𝑇 and then cool it back to the base temperature for
𝑎
imaging (Supplementary Fig. 9). The flux contrast reduces with increasing 𝑇 and disappears above 5
𝑎
K. Plotting the flux contrast with resistance as a function of temperature (Fig. 4f), we find that 𝐌 is
𝐉
roughly constant below 𝑇 = 3 K and starts to decrease appreciably above that. While the
𝑎
superconductor is still in a zero-resistance state at 4 K, which is also evident from a small but finite
superfluid density on the right domain of the sample (Fig. 4f), 𝐌 is halved upon reaching this
𝐉
temperature. At 5 K, the resistance is still an order of magnitude smaller than the normal state
resistance. And yet, 𝐌 is almost completely invisible. This suggests that the long-range ferromagnetic
𝐉
ordering of magnetic impurities is disrupted by thermal fluctuations in the underlying superconducting
condensate. Considering the manifestation of ferromagnetic order is linked to the superfluid density, the
Curie temperature is consistent with the superconducting transition temperature.

The ferromagnetic coupling between impurity moments can not only be mediated by a bias current but
also by a circulating supercurrent from the Meissner effect. After zero-field cooling (Supplementary Fig.
10a), we apply an out-of-plane magnetic field 𝐻 at base temperature much smaller than the lower
critical field of Fe(Se,Te) (~ 500 Oe) [25] to induce the Meissner current. We image the flux signal with
the field present and change the field after each scan. The flux response is diamagnetic at 𝐻 = 0.64 G
for both the top and bottom domains (Supplementary Fig. 10a). But at 𝐻 = 1.08 G the bottom part
starts to turn positive, and the paramagnetic area grows bigger and stronger with increasing field
(Supplementary Fig. 10). At 𝐻 = 1.51 G, the lower domain is completely paramagnetic, while the top
domain is still diamagnetic. Upon reducing 𝐻 = 0.64 G, however, the top domain also changes sign so
that both domains are paramagnetic. This is in clear contrast with the flux signal at the same field from
the upsweep. The positive flux signal remains after removing the field. The flux pattern after reaching
𝐻 = -1.51 G in a down-sweep and then returning to -0.6P4 G in an up-sweep is anti-symmetric with that
at 1.51 G and 0.64 G in a reversed field history, respec tively. And the remanent flux signal is negative
upon returning to zero. The whole hysteresis loop with 𝐻 is better visualized by sampling one location
within each domain (Supplementary Fig. 10b). The overall hysteresis is consistent with the 𝐌 induced
𝐉
by a bias supercurrent.

The remanent flux contrast inducedI by the Meissner current appears similar to that induced by the bias
current. This is the case only because of the specific current leads used in the bias experiment. When we
use electrodes located on tAhe same side of the sample, a large part of the supercurrent circles around the
sample (Figs. 3g-h and Supplementary Figs. 7a-b), which is similar to the current distribution of the
Meissner current (Supplementary Fig. 10a). As a result, the magnetization texture is also composed of
in-plane moments pointing towards or away from the center of the sample. However, when we use
electrodes on the opposite side of the sample, the supercurrent flows unidirectionally (Fig. 4e and
Supplementary Fig.8c). The remanent flux we obtain in this scenario is consistent with a uniform in-
plane magnetization.

Methods
Sample growth and preparation
Single crystals of FeTe Se were grown using the Bridgman method. A mixture of Fe (99.99%), Te
1-x x
(99.999%), and Se (99.999%) powders with appropriate ratios was loaded into a silica Bridgman
ampoule and sealed into a second ampoule to avoid cracking upon cooling. The ampoule was evacuated,
sealed, and heated to 920 ◦C, then slowly cooled to room temperature. The single crystals as grown with
a typical size of 5.0 × 5.0 × 1.0 mm3 can be easily cleaved perpendicular to the c axis.

The average Te:Se composition is 0.55:0.45. The inters titial Fe density in as-grown crystals is high
enough so that the superconducting volume fraction and T are both reduced [43,46]. And the sSQUID
c
susceptometry shows weak diamagnetic suscep tibility as reported earlier by Jiang et al. [42]. Annealing
in an oxygen-rich environment or vacuum may significantly decrease the interstitial Fe density and
increase the superfluid density [43-47C]. However, in this work, we focus on the superconducting
Fe(Se,Te) crystals with relatively hiIgh interstitial Fe density to facilitate the ferromagnetic interaction.

We use mechanical exfoliaAtion to obtain Fe(Se,Te) flakes for our measurements. Fig. 3a presents an
optical image of a flake and a sketch of our nano-SQUID susceptometer probe. This particular flake is
300 nm thick (Supplementary Fig. 3c) and has a lateral size of 90×30 𝜇m2. It is exfoliated from a
large single crystal about 2 mm in size using the scotch tape method and then transferred onto a SiO
2
substrate to be in contact with pre-deposited Cr/Au electrodes. After that, the Al electrodes on the
bottom edge (bright yellow in Fig. 3a) are evaporated onto the flake.

Scanning SQUID microscopy
Scanning SQUID (sSQUID) measurements were performed in a closed-cycle He-4 refrigerator. The
thermometer was placed on the sample holder. The heater for the variable-temperature measurements
shown in Fig. 2 was placed on the substrate, while the heater for those in Fig. 4 was on the sample
holder. A small temperature gradient between the sample and the thermometer is expected for the
former.

The nano-SQUID sensors used for this study were scanning 2-junction SQUID susceptometers with two
balanced pickup loops of 2 μm diameter in a gradiometric configuration, each surrounded by a one-turn
field coils of 10 μm diameter. Besides, the distance between the pick-up loop and sample also puts
restrictions to spatial resolution. The distance is usually about several micrometers determined by the
polishing of chip edge and the mounting angle of the probe, making the overall effective resolution a bit
larger than 2 μm. It is also worth noting that our measurements are capturing the field distribution
several micrometers above the edge current plane, in which Bz field contrast near the edge appears less
distinct. Therefore, the spatial resolution varies with the field source.

For the scanning SQUID measurements throughout this paper, we employed three different modes to
probe magnetic properties. Magnetometry (𝛷) of sample #2 is a direct-current measurement of flux
through the pickup loop, while 𝛷 of sample #1 is demodulated at the vibration frequency of the tuning
fork carrying the nano-SQUID. The demodulated alternate-current flux signal 𝛷 reflects the gradient
AC
of the magnetic flux along the vibration axis (z) and is proportional to the direct-current flux signal
𝛷 . Both 𝛷 and 𝛷 of sample #3 are depicted in the Supplementary Fig. 12. Susceptometry
DC DC AC
(𝑑Φ/𝑑𝐼 ) measures the inductive response of the sample through the pickup loop to a modulation
𝐹
current in the field coil 𝐼 . The magnetometry is perfo rmed simultaneously with susceptometry.
𝐹
Because the pickup loop is parallel to the sample surface, it is only sensitive to the local out-of-plane
magnetic field.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript reports on experimental detection of ferromagnetic coupling between localized magnetic moments in a\nsuperconductor with Rashba SOC. It uses scanning SQUID microscopy to detect anomalous supercurrents due to this\nferromagnetic coupling above a bias threshold which interestingly remains even when the bias is removed. The evidence of\nswitching the magnetic moments by applying a bias supercurrent above a threshold opens novel avenues for the design of\ncryogenic magnetic memories.\nThis is an interesting study which uncovers this rather unusual mechanism of ferromagnetic coupling mediated by\nsuperconductivity. Before this manuscript can be considered for acceptance, the authors to clarify the points below:\n1. From the magnetic domain imaging in Figure 2 and the corresponding extended figure, it seems that these magnetic\ndomains are several microns long and wide. It is not clear what sets this initial domain size.\n2. The explanation of aligning the spins along the centre line is also not clear. It multiple spins are present within the\ncoherence length, it seems that an antiferromagnetic coupling will reduce the overall free energy. Why is then a\nferromagnetic alignment preferred in equilibrium (no external supercurrent)?\n3. Page 4, line 144: Figure 2a is showing result from one flake and flake-to-flake variation is not clear from here. Image from\na second flake in the main figure or in the extended information is needed to show the variation. Also, the authors should\nexplain in a few sentences what the figure shows and how the signal relates to the superfluid density.\n4. Figure 2: it is not clear that why the susceptometry and magnetometry images do not exactly align with respect to the\ndotted line marking the flake boundary. It seems that the magnetometry image is outside the boundary on the right hand side\nand the susceptometry data does not go all the way to the boundary. The authors should explain this.\n5. Page 4, line 147 - the comparison image should be provided as an inset from reference 14.\n6. Page 5, first line (156) and figure 2: the magnetic contrast appear to decay faster than the superfluid density as Tc is\napproached. What is the reason for this faster disappearence of the magnetic signal?\n7. Last line of Fig. 2 caption should read, “…the magnetic domains results from…”", "reviewer_2": "Xiang et al. report on a scanning SQUID study of flakes of Fe(Se,Te) with relatively high Fe density. The main observation is\nflux patterns that change in a hysteretic way after applying a current bias through the flakes. A comprehensive study of this\nbehavior is presented, including an analysis of magnetic field training through Meissner currents. Data from three flakes are\nshown. The remnant flux is attributed to magnetization induced by a RKKY-like interaction, enabled in the superconducting\nstate by strong SOC. This would be an interesting demonstration of the interplay between magnetic impurities and\nsuperconductivity. However, there are several major issues with the interpretation of the data, which I describe below. These\nproblems preclude me from recommending publication at this point.\nMost importantly, the authors do not properly account for the inhomogeneity of the samples, which I believe is sufficient to\nexplain the observations. The susceptibility of all the flakes in this study is highly inhomogeneous. For example the top part\nof the flake in Fig. 3 does not appear to be superconducting even at the base temperature. This means the critical current is\ndifferent for different regions in the sample. The change to the current distribution, and the remnant magnetization, are\ncorrelated with the top part of that flake turning normal. This motivates the following interpretation: once the critical current for\nthe top part is reached, the current flows through the edges of the bottom part, generating a magnetic field through the flake.\nWhen the current is gradually removed, the top part turns superconducting in the presence of this field. This results in\nMeissner currents and/or vortices, and because the Pearl length is so large, they manifest as a “homogeneous” flux signal.\nThe other data shown are consistent with this alternative interpretation, for example the remnant flux pattern in extended\ndata 11 is correlated with the superfluid density landscape, rather than with current density pattern that would be generated\nin this geometry.\nTo exclude this explanation, I suggest the authors either repeat the experiment on a single crystal sample, or on more\nhomogeneous flakes, and provide a better characterization of the local critical temperature and current density distributions\nin the existing data.\nAdditional issues are listed below:\n1) An image of the vortex distribution after cooling down in the presence of a magnetic field is missing. This would be helpful\nin order to compare with the magnetic landscape reported, and rule out vortices.\n2) What is the spatial resolution in these measurements? The authors say the pick-up loop diameter was 2 um, but they can’t\nresolve the electrodes in extended data figure d,e.\n3) The range shown in most images is insufficient to understand the current flow or magnetization. Magnetic field lines that\ngo out of plane and generate a positive signal should eventually close, generating a negative signal somewhere else. The\nauthors should measure further away from the flakes to capture the complete signal.\n4) Similarly, the authors argue that the remnant flux signal is due to edge currents, but the data do not resemble the field\ngenerated according to Biot-Savart law. Can the authors explain this?\n5) The dashed lines in Figure 3 (and the corresponding extended data figures) are highly misleading. The optical image\nclearly shows the flake extends to the right of where the dashed line is marked, and the susceptibility image in Fig. 3b shows\nthat portions of this area are superconducting. Yet the remnant flux appears only within the dashed lines. The authors should\nexplain this and provide imaging of the full device.\n6) The streamlines overlaid on Fig. 3f,h,k are confusing. Firstly, they create a false impression that the current flows\nhomogeneously along the streamlines. The authors should try a different way to visualize this, perhaps by making the\nthickness of the lines proportional to the magnitude of the current density. Secondly, in panels h and k, there are additional\nstreamlines overlaid. It is not clarified in the text how these lines were generated, and I do not see a way to divide the current\npattern into two origins just from the experimental data.\n7) In extended data fig. 6, the current appears to flow at the edge of the sample, even though Ic was exceeded. Can the\nauthors explain that? The authors should compare this with images of the current flow taken above the transition\ntemperature, to ensure the sample is completely normal."}, "author_response": "Response to Reviewer #1\nThe manuscript reports on experimental detection of ferromagnetic coupling between\nlocalized magnetic moments in a superconductor with Rashba SOC. It uses scanning\nSQUID microscopy to detect anomalous supercurrents due to this ferromagnetic coupling\nabove a bias threshold which interestingly remains even when the bias is removed. The\nevidence of switching the magnetic moments by applying a bias supercurrent above a\nthreshold opens novel avenues for the design of cryogenic magnetic memories.\nThis is an interesting study which uncovers this rather unusual mechanism of\nferromagnetic coupling mediated by superconductivity.\n-We thank the Reviewer #1 for concisely summarizing our work and highlighting its\nimportance. We are also grateful for the valuable comments that help us to improve it.\nWe have addressed the specific questions and comments below.\nBefore this manuscript can be considered for acceptance, the authors to clarify the points\nbelow:\n1. From the magnetic domain imaging in Figure 2 and the corresponding extended figure,\nit seems that these magnetic domains are several microns long and wide. It is not clear\nwhat sets this initial domain size.\n-We thank the reviewer for bringing up this issue. The initial domain size might be\ndetermined by the London penetration depth, coherence length, average distance\nbetween interstitial iron atoms and fluctuations during the cooling process. Actually, it\nmay be smaller than the spatial resolution of our probe. We have added this discussion\nto Lines 166-169 in the revised manuscript.\n2. The explanation of aligning the spins along the centre line is also not clear. It multiple\nspins are present within the coherence length, it seems that an antiferromagnetic coupling\nwill reduce the overall free energy. Why is then a ferromagnetic alignment preferred in\nequilibrium (no external supercurrent)?\n-Even without an external supercurrent, there is spontaneous anomalous supercurrent\nexcited around interstitial iron atoms. Taking London-Pearl screening effects into\nconsideration, the interaction between them may lead to ferromagnetic coupling under\ncertain conditions [Phys. Rev. B 108, L180506 (2023)]. Indeed, the ferromagnetic order\nis the one that minimized the energy of the anomalous currents induced by the\nmagnetoelectric effect, due to the SOC, around each impurity. Consequently, the ground\nstate is a domain state and the uniform ferromagnetic alignment can be facilitated by\nexternal bias current. We have added this clarification to Lines 100-104.\n3. Page 4, line 144: Figure 2a is showing result from one flake and flake-to-flake variation\nis not clear from here. Image from a second flake in the main figure or in the extended\ninformation is needed to show the variation. Also, the authors should explain in a few\nsentences what the figure shows and how the signal relates to the superfluid density.\n-We thank the reviewer for this nice suggestion. In the revised manuscript, we have\nassigned a specific number for each flake (#1-#3) to show the variation. Also, we have\nadded some explanations for Figure 2a to Lines 147-151:‘We apply an alternating\nmagnetic field using the field coil on SQUID chip and demodulate the magnetic flux\nchanges on the sample to obtain the local magnetic susceptibility (Figure 2a). The\nnegative susceptibility is contributed by Meissner supercurrent directly related to the\nsuperfluid density.’\n4. Figure 2: it is not clear that why the susceptometry and magnetometry images do not\nexactly align with respect to the dotted line marking the flake boundary. It seems that the\nmagnetometry image is outside the boundary on the right hand side and the susceptometry\ndata does not go all the way to the boundary. The authors should explain this.\n-We thank the reviewer for pointing out this issue. In practice, we delineate the sample\ncontours based on the local susceptibility image. Due to the finite spatial resolution, the\nboundary transitions are characterized by a gradual decay rather than abrupt changes,\nintroducing some uncertainty to the contours. We have made slight adjustments to the\nselection of the contour in Figure 2 to make it visually more consistent with the\nmagnetic susceptibility image presented under the current colormap. As for the\nmagnetic contrast outside the boundary, it might come from magnetic moments and\nsupercurrent near the edge which has a finite out-of-plane magnetic field outside the\nboundary of the sample. We have added these explanations to Line 161-164.\n5. Page 4, line 147 - the comparison image should be provided as an inset from reference\n14.\n-We thank the reviewer for this suggestion. We have provided a comparison image from\nreference 14 ([Phys. Rev. X 13, 011046 (2023)]) as an inset to Figure 2a.\n6. Page 5, first line (156) and figure 2: the magnetic contrast appear to decay faster than\nthe superfluid density as Tc is approached. What is the reason for this faster disappearence\nof the magnetic signal?\n-The magnetic contrast depends on superfluid density in two respects. First, the\nsuperfluid is the mediator for the ferromagnetic coupling between magnetic impurities.\nSecond, the magnetization generates anomalous supercurrent due to the magneto-\nelectric effect. And the magnetic contrast we observed is from the anomalous\nsupercurrent. The reduction of superfluid density thus has a compound effect on the\nmagnetic contrast we detect, making it decreasing faster as Tc is approached. We have\nadded this discussion to Line 177-183.\n7. Last line of Fig. 2 caption should read, “…the magnetic domains results from…”\n-We thank the reviewer for pointing out this language problem. We have modified this\nsentence in Figure 2 caption as suggested.\nResponse to Reviewer #2\nXiang et al. report on a scanning SQUID study of flakes of Fe(Se,Te) with relatively high\nFe density. The main observation is flux patterns that change in a hysteretic way after\napplying a current bias through the flakes. A comprehensive study of this behavior is\npresented, including an analysis of magnetic field training through Meissner currents. Data\nfrom three flakes are shown. The remnant flux is attributed to magnetization induced by a\nRKKY-like interaction, enabled in the superconducting state by strong SOC. This would be\nan interesting demonstration of the interplay between magnetic impurities and\nsuperconductivity.\n-We thank the reviewer for the careful reading and high evaluation of our manuscript.\nWe also appreciate the constructive suggestions and the opportunity to improve it.\nHowever, there are several major issues with the interpretation of the data, which I describe\nbelow. These problems preclude me from recommending publication at this point.\nMost importantly, the authors do not properly account for the inhomogeneity of the samples,\nwhich I believe is sufficient to explain the observations. The susceptibility of all the flakes\nin this study is highly inhomogeneous. For example the top part of the flake in Fig. 3 does\nnot appear to be superconducting even at the base temperature. This means the critical\ncurrent is different for different regions in the sample. The change to the current distribution,\nand the remnant magnetization, are correlated with the top part of that flake turning normal.\nThis motivates the following interpretation: once the critical current for the top part is\nreached, the current flows through the edges of the bottom part, generating a magnetic\nfield through the flake. When the current is gradually removed, the top part turns\nsuperconducting in the presence of this field. This results in Meissner currents and/or\nvortices, and because the Pearl length is so large, they manifest as a “homogeneous”\nflux signal.\nThe other data shown are consistent with this alternative interpretation, for example the\nremnant flux pattern in extended data 11 is correlated with the superfluid density landscape,\nrather than with current density pattern that would be generated in this geometry.\nTo exclude this explanation, I suggest the authors either repeat the experiment on a single\ncrystal sample, or on more homogeneous flakes, and provide a better characterization of\nthe local critical temperature and current density distributions in the existing data.\n- We thank the reviewer for bringing up an alternative interpretation of our findings. We\nacknowledge the inhomogeneity of the samples, which arises from the stochastic\ndistribution of interstitial iron atoms. However, our data is not consistent with the\nvortex\ntrapping scenario, as we elucidate below.\n1. In Extended Data Figures 8b and e, we use two electrodes on the bottom edge of the\nsample so that the bias current favors to flow along the bottom edge. This current\nconfiguration generates a positive magnetic field within the sample. However, we\nobserved remanent flux signal with a negative polarity (Extended Data Figures 8b),\nwhich contradicts the vortex trapping scenario. The same argument also applies to the\nremnant flux pattern depicted in Extended Data Figure 8c, which is more consistent\nwith the anomalous supercurrent picture we propose [Phys. Rev. B 102, 060506, (2020);\nPhys. Rev. B 108, L180506 (2023)].\n2. We have expanded the image range of sample #2 in Extended Data Figures 4 and 7.\nWe can see that the superfluid density on the left part of the flake is depleted at I = 200\nb\nμA. Hence, this region would be expected to be more susceptible to vortex trapping.\nHowever, this region shows negligibly small remnant flux.\n3. Our observation of the sudden onset of edge current above some threshold does not\nagree with vortex trapping. From Extended Data Figures 4d, we can see that the\ndiamagnetism of the sample at I = 200 μA is still present on the bottom edge, which is\nb\nthe region where the edge current shows a strong contrast (Fig. 3).\n4. The temperature dependence of the remnant flux (Fig. 4f) provides additional\nevidence against vortex penetration. The variation of integrated magnetic flux with\nincreasing temperature is much smaller than the flux quantum, which is inconsistent\nwith leakage of vortices with increasing temperature.\n5. We have repeated the experiment on a more homogeneous flake sample (Extended\nData Figure 12). We have observed the spontaneous magnetic domains (Extended Data\nFigure 12b) as well as the alignment of domains by a bias current (Extended Data\nFigures 12c and d). In this sample, the switching current is about 1.6 mA (Extended\nData Figure 12f) which is far below the critical current of 6 mA (Extended Data Figure\n12e), making the possibility of vortex trapping highly unlikely. Furthermore, the sign\nof the remanent flux is consistent with that on samples #1 and #2 and opposite from the\nBiot-Savart field of the current.\nWe have added these discussions to Lines 216-221 and Lines 304-311 in the revised\nmanuscript.\nAdditional issues are listed below:\n1) An image of the vortex distribution after cooling down in the presence of a magnetic field\nis missing. This would be helpful in order to compare with the magnetic landscape reported,\nand rule out vortices.\n-We are grateful for the reviewer's constructive suggestion. In the revised manuscript,\nwe have accordingly incorporated the magnetic landscape of flake #1 after field cooling\nto Extended Data Figure 2. Different from the Fe(Se,Te) flake in the low interstitial Fe\ndensity regime [Phys. Rev. X 13, 011046 (2023)], it always shows magnetic domains\nrather than vortices, even if the applied field is strong enough to induce vortex\npenetration. It suggests that in the high interstitial Fe density regime, ferromagnetic\nstructure is a more favorable state for magnetic impurities due to the Rashba coupling.\nWe have added these discussions to Lines 169-173.\n2) What is the spatial resolution in these measurements? The authors say the pick-up loop diameter was 2 um, but they can’t\nresolve the electrodes in extended data figure d,e.\n-The pick-up loop diameter is 2 μm which limits the optimal spatial resolution. Besides,\nthe distance between the pick-up loop and sample is also a restriction. The distance is\nusually about several micrometers determined by the polishing of chip edge and the\nmounting angle of the probe, making the overall resolution a bit worse than 2 μm.\nWe can resolve the electrodes through the boundaries of positive and negative current\nflux signals in Extended Data Figure 8f, attesting to the resolution. However, such a\nfeature is inherently absent when the bias is on the same side of the sample (Extended\nData Figures 8d-e) because the current flux has the same sign in the regions between\nthe two electrodes. We have made modifications to Lines 457-461 in the revised\nmanuscript.\n3) The range shown in most images is insufficient to understand the current flow or\nmagnetization. Magnetic field lines that go out of plane and generate a positive signal\nshould eventually close, generating a negative signal somewhere else. The authors should\nmeasure further away from the flakes to capture the complete signal.\n-We appreciate the reviewer’s helpful advices. We have updated Extended Data Figure\n7 to provide a comprehensive visualization of the complete current flow. Extended Data\nFigure 4 has also been modified to show susceptibility and remnant flux across the\nentire device.\nThe reviewer is totally correct about the flux line having to close. However, there is\nlimitless space outside the sample and therefore the return flux typically extends over a\nmuch larger area than the sample. This is the reason that the return flux signal outside\nthe sample boundary is significantly weaker than the flux signal within the boundary.\nIn order to enhance the signal contrast, we utilize AC measurements that are\ndemodulated at the vibration frequency of the tuning fork carrying the nano-SQUID.\nSince such a demodulated flux signal is more sensitive to the gradient of the magnetic\nfield, this method allows for a clearer visualization of the flux signal of opposite sign\nacross the sample boundary (Extended Data Figures 12c and d).\n4) Similarly, the authors argue that the remnant flux signal is due to edge currents, but the\ndata do not resemble the field generated according to Biot-Savart law. Can the authors\nexplain this?\n- Due to the geometry constraint of the planer nano-SQUID, we are capturing the field\ndistribution several micrometers above the edge current plane, in which Bz field\ncontrast near the edge appears less distinct. We have added these supplementary notes\nto Lines 461-464.\nWhen measuring with the AC flux mode discussed above, we can see clear edge flux\nreversal in consistent with edge current according to Biot-Savart law (Extended Data\nFigures 12c and d).\n5) The dashed lines in Figure 3 (and the corresponding extended data figures) are highly\nmisleading. The optical image clearly shows the flake extends to the right of where the\ndashed line is marked, and the susceptibility image in Fig. 3b shows that portions of this\narea are superconducting. Yet the remnant flux appears only within the dashed lines. The\nauthors should explain this and provide imaging of the full device.\n-We thank the reviewer for bringing up this issue. We have modified Extended Data\nFigure 4 and Extended Data Figure 7 to show the magnetometry and susceptometry\nimages of the full device.\nThe dashed lines are introduced according to the susceptometry image (Extended Data\nFig. 4d). As explained in more detail in the revised manuscript Lines 192-194, the\nsegregation of diamagnetic patches is related to inhomogeneous interstitial Fe density.\nWe find the remnant flux is predominantly located within the dashed circle probably\ndue to its higher superfluid density, which correlates with the remnant flux intensity.\n6) The streamlines overlaid on Fig. 3f,h,k are confusing. Firstly, they create a false\nimpression that the current flows homogeneously along the streamlines. The authors\nshould try a different way to visualize this, perhaps by making the thickness of the lines\nproportional to the magnitude of the current density. Secondly, in panels h and k, there are\nadditional streamlines overlaid. It is not clarified in the text how these lines were generated,\nand I do not see a way to divide the current pattern into two origins just from the\nexperimental data.\n-We thank the reviewer for the very constructive suggestions. We have revised the\ncurrent streamlines in Figs. 3f,h,k and made the thickness of the lines proportional to\nthe magnitude of the current density. Additional streamlines in panels h and k are\nanomalous supercurrent 𝑱 with opposite directions in the bulk (dark green lines) and\na\nalong the edges (light green lines). Our interpretation is based on the previous\ntheoretical works [Phys. Rev. B 102, 060506, (2020); Phys. Rev. B 108, L180506\n(2023)]. This model is supported by the emergence of the edge current and the remnant\nflux pattern under different current configurations. We have emphasized these\nexplanations on Lines 293-297.\n7) In extended data fig. 6, the current appears to flow at the edge of the sample, even\nthough Ic was exceeded. Can the authors explain that? The authors should compare this\nwith images of the current flow taken above the transition temperature, to ensure the\nsample is completely normal.\n-We thank the reviewer for the instructive comments. We have made clearer definition\nof critical current in the revised manuscript. At I = 260 μA, the sample is in the mixed\nb\nstate: it exhibits reduced superfluid density, but it has not reached a complete normal\nstate despite the presence of a voltage across it. The current still flows at the bottom\nedge due to the stronger superfluid density there (Extended Data Fig. 7c). As a\ncomparison, current does flow uniformly across the sample when it is completely\nnormal above the transition temperature [Science Bulletin 66, 425-432 (2021)]. These\nresults suggest that the edge remnant flux and anomalous current do not arise from the\nloss of superconductivity. We have added these explanatory sentences to Lines 227-232\nin the revised manuscript.\nMajor changes in the revised manuscript:\n1. Images of direct-current measurement of flux for sample #3 are added to\nExtended Data Figure 12. Explanation of two measurement modes is added to\nLines 483-489.\n2. Analysis of the remanent magnetic flux signal amplitude versus\nsuperconductivity of sample #3 compared to sample #2 is added in Lines 225-\n229, supporting the scenario that superfluid density mediates ferromagnetic\ncoupling.\n3. Further discussion of how magnetic domains prevent vortex penetration are\nadded to Lines 171-176.\n4. Explanations of Pearl vortex and its spatial distribution are added to Lines\n236-245. References related to Pearl length, spatial distribution of Pearl vortex and\nits experimental measurements are inserted to the revised manuscript.\n5. Additional illustration of current distribution with bias current is added to Lines\n313-320."}, {"round": 2, "reviewer_comments": {"reviewer_2": "The authors provide additional data and discussion, which alleviate some of my concerns.\nThe only truly convincing support for the scenario that they propose is the new extended data fig. 12, which is from a new\nflake that is more homogeneous, and where the switching current is far below the critical current. There are two points that\nmust be addressed about these new data:\n(1) To complete the picture, the authors should also include images of the current density in the superconducting and normal\nstate. Particularly, with a bias of 2mA it is critical to show that the current flows entirely on the edge (as expected if the\nsample is indeed SC) and not through some complicated path in the sample.\n(2) The critical current of this new sample (6mA) is ~30 times larger than the other devices (~200 uA). The susceptibility\nsignal also seems to be ~10 times larger and yet the magnetic signals are 10 times smaller. It appears to contradict the\nscenario put forth by the authors: if the superfluid density mediates the ferromagnetic interaction, I would expect stronger\nsuperconductivity to also give rise to stronger magnetism.\nThe authors claim that vortices never enter the samples, even when cooled with a\nsufficiently large external field. This should be explained further. In addition, the authors should provide the (London)\npenetration depth of these samples, preferably extracted from the susceptibility measurements, and compare this with the size\nof domains, size of the flakes and the spatial resolution. I am not convinced that the vortices are not simply too large to be\nresolved here.\nTo conclude, additional data and analysis are still required to fully support the interpretation\nsuggested by the authors."}, "author_response": "Response to Reviewer #2\nThe authors provide additional data and discussion, which alleviate some of my concerns.\nThe only truly convincing support for the scenario that they propose is the new extended\ndata fig. 12, which is from a new flake that is more homogeneous, and where the switching\ncurrent is far below the critical current.\n-We are grateful to the reviewer for approving the improvement of the revised\nmanuscript with the additional experimental data and discussions. As highlighted in\nExtended Data Figure 12, the new flake with improved homogeneity exhibits that the\nswitching current is substantially lower than the critical current, thereby ruling out\nalternative explanations such as the vortex-trapping scenario.\nThere are two points that must be addressed about these new data:\n(1) To complete the picture, the authors should also include images of the current density\nin the superconducting and normal state. Particularly, with a bias of 2mA it is critical to\nshow that the current flows entirely on the edge (as expected if the sample is indeed SC)\nand not through some complicated path in the sample.\n-We thank the reviewer for raising this important point. We would like to clarify that\nthe current does not flow entirely along the edge when a bias current is applied. Instead,\ncurrent distribution involves a combination of anomalous edge and bulk supercurrents\n(Figs. 2g and j), which are induced by the long-range in-plane magnetization intrinsic\nto the Rashba superconductor. It is only after the bias current is removed can we observe\nthe anomalous supercurrent which flows only along the edge. For Sample #3,\nregretfully we were unable to perform imaging at the high bias current of 2 mA due to\nsome technical difficulties (the heating under such high current from resistive part of\nthe current leads induces large noise in the nano-SQUID). However, the flux images\nunder an applied current are presented for sample #2 in Figure 3 and 4 of the manuscript,\nwhich is consistent with the analysis above.\nThe reviewer's concern appears to center on the vortex penetration effects. We\nemphasize that superconductors with comparable Pearl lengths (~100 μm) exhibit well-\nestablished vortex signatures as demonstrated in previous studies [Appl. Phys. Lett. 5,\n65-66, (1964); Phys. Rev. Lett. 92, 157006, (2004); Nat. Commun. 16, 2696, (2025)].\nSpecifically, vortex-induced magnetic profiles manifest characteristic 1/r (r being the\ndistance from vortex core) spatial dependence within the Pearl length regime.\nImportantly, homogenous remanent flux distribution here is inconsistent with vortex-\ninduced magnetic profile and strongly supports our interpretation of circulating current.\nWe have substantially expanded our discussion of these points into the revised\nmanuscript in Lines 313-320 and Lines 236-245.\n(2) The critical current of this new sample (6mA) is ~30 times larger than the other devices\n(~200 uA). The susceptibility signal also seems to be ~10 times larger and yet the magnetic\nsignals are 10 times smaller. It appears to contradict the scenario put forth by the authors:\nif the superfluid density mediates the ferromagnetic interaction, I would expect stronger\nsuperconductivity to also give rise to stronger magnetism.\n-We thank the reviewer for catching this detail. The magnetic images shown for the two\nsamples are from different measurement modes and therefore cannot be compared\ndirectly. The flux images in Extended Data Figure 12 employ lock-in detection at the\ntuning fork frequency (AC demodulation) to enhance sensitivity, whereas Fig. 3\npresents DC flux measurements of sample #2.\nIn order to quantitative compare the remanent flux of the two samples, we have included\nDC flux data for sample #3 in the revised Extended Data Figure 12. The DC flux\nmagnitude of sample #3 is indeed about 10 times larger than sample #2, in consistent\nwith their relative diamagnetic susceptibility. This further supports our scenario that\nsuperfluid density mediates the ferromagnetic interaction. We have explicitly addressed\nthis quantitative detail in Lines 483-489 and added analysis in Lines 225-229.\nThe authors claim that vortices never enter the samples, even when cooled with a\nsufficiently large external field. This should be explained further.\n-We thank the reviewer for bringing up this point. We argue that the presence of in-\nplane magnetization in our samples may compete with vortex formation. Previous\nworks [14] showed that the magnetic impurity moment in Fe(Se,Te) tend to align with\nthe magnetic field of the vortices (Fig. 1a). Therefore, a long-range in-plane\nmagnetization of impurity moments disfavors vortices (which are phase windings)\nnucleating in the sample (Fig. 1d). This argument is also corroborated by recent\ncalculations taking into account the London-Pearl screening effects [10]. We have\nadded these explanations to the revised manuscript in Lines 171-176.\nIn addition, the authors should provide the (London) penetration depth of these samples,\npreferably extracted from the susceptibility measurements, and compare this with the size\nof domains, size of the flakes and the spatial resolution. I am not convinced that the vortices\nare not simply too large to be resolved here.\n-We thank the reviewer for the insightful suggestion. We would like to clarify the\nconcept of penetration depth involved in our manuscript. The London penetration depth\n(𝜆) describes the exponential decay of the magnetic field near the surface of a strong\nbulk superconductor, characterizing the Meissner screening effect. While in two-\ndimensional superconductors or superconductors with low superfluid density defined\nby 𝜆 ≫ 𝑑 , where 𝑑 is the thickness of the sample, Pearl length (Λ ) governs the\nP\nlateral spread of magnetic flux from vortices. The relationship between these two\nmagnetic length scales is given by Λ = 2𝜆2/𝑑 [Appl. Phys. Lett. 5, 65-66, (1964)].\nP\nFor sample #1, susceptibility measurement (Extended Data Figure 1) yields Λ ≈\nP\n87 μm . Sample #3 exhibits similar susceptibility and Λ , while sample #2 shows\nP\nweaker susceptibility, implying a larger Λ . London penetration depth 𝜆 of these\nP\nsamples derived from the aforementioned relationship is several micrometers. The size\nof the observed magnetic domains (Extended Data Figure 1 and Extended Data Figure\n12) is also on the order of several micrometers, which is close to the spatial resolution\nof our probe.\nFor all three samples, the Pearl length Λ exceeds the physical dimensions of the\nP\nflakes. If vortices were present, the flux profile would decay as 1/Λ 𝑟 near the vortex\nP\ncore and as Λ /𝑟3 for distance greater than Λ , where 𝑟 is the distance from the\nP P\nvortex center [Appl. Phys. Lett. 5, 65-66, (1964); Phys. Rev. Lett. 92, 157006, (2004);\nNat. Commun. 16, 2696, (2025)]. However, the remanent flux pattern does not follow\nsuch a decay law, indicating inconsistency with the explanation of vortex-penetration.\nWe have incorporated these analyses into the revised manuscript in Lines 236-245.\nTo conclude, additional data and analysis are still required to fully support the interpretation\nsuggested by the authors.\n-We sincerely thank the reviewer for the constructive suggestions that help us further\nimprove our manuscript. In the revised version, we have incorporated additional data\nand analysis to further exclude vortex-related effect and support the scenario of\nsuperfluid density mediated ferromagnetic coupling. We appreciate the reviewer's time\nand feedback, and we trust the revised manuscript now meets the required standards."}, {"round": 3, "reviewer_comments": {"reviewer_2": "I would like to reiterate that the experimental data presented are comprehensive, and that\nthe interpretation offered by the author is consistent with the data. However, the authors failed to convincingly reject the trivial scenario where inhomogeneity\nand vortex penetration lead to the observed signals. Therefore, I find myself unable to recommend the publication without at\nleast a serious discussion and consideration of all other alternative interpretations.\nThe penetration of vortices in these samples is complicated. The inhomogeneity of the samples, combined with the “training”\nschemes that are used, where the current and magnetic field distributions in the sample are inhomogeneous, allow a non-\nuniform density of vortices to penetrate the sample. The density of these vortices may be too high to resolve individually with\nthe 2 um resolution of the SQUID sensor. Therefore the discussion of the Pearl length and signature of an individual vortex\nis irrelevant. The only convincing approach is experimental, that is to show that a current bias applied to a homogeneous sample (1)\ndoes not suppress superconductivity locally, and (2) leads to a remnant magnetization. This evidence is missing and therefore the vortex penetration cannot be rejected.", "reviewer_3": "This work imaged magnetic flux distribution using scanning SQUID microscopy and claimed supercurrent-controlled\nferromagnetism in a spin–orbit-coupled superconductor with magnetic impurities. Overall, this claim is highly questionable,\nand the authors made several major claims without providing reliable scientific evidence. I outline some key issues below:\n1.\nIt is risky to claim ferromagnetism solely based on inhomogeneous magnetic field distribution. While ferromagnetism or\nferromagnetic domains produce inhomogeneous magnetic fields, such distributions can also arise from many other sources,\nincluding ferrimagnetism, inhomogeneous superconductivity, surface roughness or friction-induced artifacts, poor-quality\nsamples with inhomogeneous clusters, and extrinsic magnetic contamination. If ferromagnetism truly exists, it can be easily\nproved through transport, magnetization, or optical measurements. The authors do not provide any such convincing\nexperimental support.\n2.\nAlthough the authors claim ferromagnetism, they do not convincingly identify its key characteristics, such as domain size,\ncoercive field, or Curie temperature (Tc). What additional experimental evidence supports these characteristics? In\nparticular, Tc and coercive field are extremely straightforward to measure, yet the authors provide no such data.\n3.\nIt is surprising that the authors did not pattern their samples into a regular shape. Instead, they reached their central claims\nbased on randomly distributed current flow in the samples (the authors appear uncertain about the homogeneity of their\nsamples in terms of both structure and resistivity).\n4.\nMoreover, the authors’ claim of in-plane ferromagnetism contradicts their observed magnetic field distribution. If the\nmagnetization were truely in-plane, the positive and negative magnetic flux should be roughly balanced, especially since the\nassumed “magnetic domains” in the authors’ samples appear to be on the micrometer scale based on the scanning SQUID\npatterns.\n5.\nFinally, what if the magnetic impurities are not intrinsic? They could simply be magnetic contaminants introduced during\nsample preparation or growth. If that is the case, investigation of this phenomenon would be meaningless.", "reviewer_4": "The paper entitled, \"Supercurrent-controlled spontaneous ferromagnetism of magnetic impurities in a spin-orbit-coupled\nsuperconductor\" describes scanning SQUID measurements showing evidence for a long-range ferromagnetic order of the\nimpurity moments on the surface of Fe(Se,Te).\nThe presented measurements are of high quality and their interpretation and analysis appear appropriate. In particular, the\nauthors do a careful job checking various alternative scenarios that may explain their magnetic signal, including vortex\ntrapping. A lengthy supplementary section contains useful information in this respect. Altogether, I find the data and analysis\nconvincing and interesting.\nThe scanning probe aspect of the work is expertly done and is a good example of how scanning SQUID microscopy can be\nused to reveal the microscopic mechanisms behind superconducting phenomena in ways that conventional measurements\nwithout spatial resolution cannot.\nThe authors have also satisfactorily addressed the detailed comments of reviewer #2, in particular his/her questions relating\nto the possibility that the observations can be explained by vortex entry.\nFor these reasons, I recommend that the manuscript be published after the address one small concern, discussed below.\n1) I did not grasp the difference between the data shown in Fig. 2b from sample #1. Here the sample is measured in the\nvirgin state at 1.68 K. Magnetic domains are observed. However, in Fig. 3d, after the same zero-field cooling procedure the\nsample is again in the virgin state, but shows no contrast at all and no magnetic domains. What is the difference here? Why do\nthe samples behave differently? Did I miss something? Could this be made clearer in the text?"}, "author_response": "Response to Reviewer #2\nI would like to reiterate that the experimental data presented are comprehensive, and that\nthe interpretation offered by the author is consistent with the data.\n-We sincerely appreciate the reviewer's acknowledgment of the comprehensiveness of\nour experimental data and the consistency of our interpretation. This positive\nfeedback reinforces our confidence in the robustness of our findings. We have\ncarefully considered all reviewer’s comments and have made revisions to further\nenhance the clarity and depth of our manuscript.\nHowever, the authors failed to convincingly reject the trivial scenario where inhomogeneity\nand vortex penetration lead to the observed signals. Therefore, I find myself unable to\nrecommend the publication without at least a serious discussion and consideration of all\nother alternative interpretations. The penetration of vortices in these samples is\ncomplicated. The inhomogeneity of the samples, combined with the “training”\nschemes that are used, where the current and magnetic field distributions in the sample are\ninhomogeneous, allow a non-uniform density of vortices to penetrate the sample. The\ndensity of these vortices may be too high to resolve individually with the 2 um resolution of\nthe SQUID sensor. Therefore the discussion of the Pearl length and signature of an\nindividual vortex is irrelevant. The only convincing approach is experimental, that is to show\nthat a current bias applied to a homogeneous sample (1) does not suppress\nsuperconductivity locally, and (2) leads to a remnant magnetization. This evidence is\nmissing and therefore the vortex penetration cannot be rejected.\n-We are grateful to the reviewer's insightful comment and proposal for excluding\nvortex penetration scenario. We have followed the proposal to perform additional\nexperiments, which are included as a major change in the revised manuscript.\nTo summarize the results briefly, we have identified a sample exhibiting uniform\ndiamagnetism and subjected it to a current bias below its critical current (Extended\nData Fig. 13). Importantly, no local suppression of superconductivity was observed,\nas evidenced by the absence of reduction in diamagnetism when the bias current is\napplied (Extended Data Figs. 13d and g). Upon removal of the current, the sample\nexhibited a remnant magnetization (Extended Data Figs. 13f and i). These\nexperimental results provide strong evidence against the trivial scenario of vortex\ntrapping. We have included these data and the analysis in the revised manuscript in\nLines 230-234 to offer a more comprehensive discussion.\nWe believe these additional experiments and the revised manuscript adequately\naddress the reviewer's remaining concerns.\nResponse to Reviewer #3\nThis work imaged magnetic flux distribution using scanning SQUID microscopy and\nclaimed supercurrent-controlled ferromagnetism in a spin–orbit-coupled superconductor\nwith magnetic impurities. Overall, this claim is highly questionable, and the authors made\nseveral major claims without providing reliable scientific evidence. I outline some key\nissues below:\n1. It is risky to claim ferromagnetism solely based on inhomogeneous magnetic field\ndistribution. While ferromagnetism or ferromagnetic domains produce inhomogeneous\nmagnetic fields, such distributions can also arise from many other sources, including\nferrimagnetism, inhomogeneous superconductivity, surface roughness or friction-induced\nartifacts, poor-quality samples with inhomogeneous clusters, and extrinsic magnetic\ncontamination. If ferromagnetism truly exists, it can be easily proved through transport,\nmagnetization, or optical measurements. The authors do not provide any such convincing\nexperimental support.\n-We thank the reviewer for the critical assessment, which provides us with an\nopportunity to strengthen the evidence for our claims. We agree that an\ninhomogeneous magnetic field alone is not conclusive proof of ferromagnetism and\nthat alternative origins must be rigorously excluded. Our claim is not based solely on\nthe magnetic morphology, but on its unique interplay with the superconducting state\nand its response to experimental manipulation, which collectively rule out the trivial\nscenarios the reviewer has listed.\nFirstly, we acknowledge that surface roughness or friction-induced artifacts, poor-\nquality samples with inhomogeneous clusters, and extrinsic magnetic contamination\ncould produce a static magnetic contrast, yet irrelevant to superconductivity. Notably,\nas demonstrated in Fig. 2 and Extended Data Fig. 1, the magnetic domain pattern is\nnot static. The emergence of the magnetic contrast is directly tied to the superfluid\ndensity and changes randomly after each thermal cycle across the superconducting\ntransition temperature (T ). This stochastic reorganization is inconsistent with the\nc\nfixed magnetic signature expected from the trivial factors mentioned.\nSecondly, in previous versions of the manuscript, we have already shown remanent\nmagnetization and hysteresis loop as key signatures of ferromagnetism (Figs. 3 and 4,\nExtended Data Figs. 12 and 13). In the current version, we further examine a\nhomogeneous sample in which superconductivity was not locally suppressed by the\nbias current, yet a remnant magnetic flux persisted after bias removal (Extended Data\nFig. 13). These results exclude the possibilities of inhomogeneous superconductivity\nor vortex penetration as alternative explanations. We have incorporated additional\ndiscussions into the revised manuscript in Lines 167-170 and Lines 230-234.\nIn terms of the other techniques the referee mentioned, we do not agree they provide\nmore direct evidence than sSQUID. Just as we have discussed in the past and current\nversions of the manuscript (Line 116), since the sample is in the superconducting\nstate, no magnetic hysteresis is observable in transport measurements (Also see Ref.\n20 [Nature Communications 16, 7305 (2025)] for an example). For magnetization, we\nassume the referee is referring to bulk magnetometry such as MPMS measurements\nbecause the flux signal of sSQUID is also one type of magnetization-sensing\ntechnique. In fact, MPMS also employs a SQUID to readout the magnetization.\nHowever, due to the large sensing distance, such bulk magnetometry is typically not\nsensitive enough to measure surface magnetization. As for the optical technique, not\nonly is it susceptible to other magnetic or even non-magnetic sources, it is also less\nsensitive than sSQUID.\n2. Although the authors claim ferromagnetism, they do not convincingly identify its key\ncharacteristics, such as domain size, coercive field, or Curie temperature (Tc). What\nadditional experimental evidence supports these characteristics? In particular, Tc and\ncoercive field are extremely straightforward to measure, yet the authors provide no such\ndata.\n-We regret that we have not been able to convey to the referee the key difference of\nthe ferromagnetism we report from conventional ones. We agree that the parameters\nthe referee mentioned are important for conventional non-superconducting magnetic\nmaterials. In our Rashba superconductor, however, the ferromagnetic order is a direct\nconsequence of the anomalous supercurrent. Therefore, its characteristics are\ninherently connected with the superconducting state and are measured through\ncorresponding proxies, which we have reported in the past and current versions of the\nmanuscript.\nThe reviewer is correct that a coercive field is a hallmark of ferromagnetism. In our\ncase, the ferromagnetic order is driven by supercurrent, acting through a mechanism\nanalogous to spin-orbit torque. The equivalent of a \"coercive field\" is thus a critical\nswitching current. This is precisely what we demonstrate in the hysteresis loops (Fig.\n4b, Extended Data Figs. 12g and 13b), where the magnetic state depends on the\nhistory of the applied current bias. Magnetic field hysteresis is also demonstrated in\nExtended Data Fig. 10, which provides additional support for our interpretation.\nAs for the Curie temperature (T ), we emphasize that the manifestation of\nc\nferromagnetic order in our system is linked to the superfluid density, which is in term\ndetermined by the superconducting transition temperature. This relationship is shown\nin Fig. 4f and Extended Data Fig. 9, further validating our claim. We have included\nthese clarifications in the revised manuscript in Lines 278-281 and Lines 353-355 to\nensure a more thorough illustration of these characteristics.\n3. It is surprising that the authors did not pattern their samples into a regular shape. Instead,\nthey reached their central claims based on randomly distributed current flow in the samples\n(the authors appear uncertain about the homogeneity of their samples in terms of both\nstructure and resistivity).\n-We thank the reviewer for the comment. The samples in this study were\nmechanically exfoliated flakes via the standard tape-peeling method, which inherently\nproduces irregular flake shapes, just like in most studies reported on 2D materials. In\naddition, the concentration of iron atoms must be precisely optimized—a very low\ndensity is not enough to establish magnetic correlations, while a very high density\nsuppresses superconductivity. These challenges are our main concerns in sample\npreparation at the current stage. Furthermore, the shape of the sample has no bearing\non the validity of our key observations or conclusions.\nRegarding sample homogeneity, we have also prepared more uniform samples, which\nsuccessfully reproduce our central results (Extended Data Fig. 13). It is noteworthy\nthat, leveraging the unique advantages of the scanning SQUID microscopy, we can\ndirectly and clearly visualize the distribution of superconducting inhomogeneities in\nthe samples from their magnetic susceptibility—a capability that is challenging for the\naforementioned characterization techniques. We have added a sentence as an outlook\nfor future work in the revised manuscript in Lines 418-419.\n4. Moreover, the authors’ claim of in-plane ferromagnetism contradicts their observed\nmagnetic field distribution. If the magnetization were truely in-plane, the positive and\nnegative magnetic flux should be roughly balanced, especially since the assumed\n“magnetic domains” in the authors’ samples appear to be on the micrometer scale based\non the scanning SQUID patterns.\n-As we have discussed extensively in paragraphs starting on Line 297, the remanent\nmagnetic contrast we measure after the bias current is removed is from the anomalous\nedge supercurrent of a Rashba superconductor, rather than the in-plane magnetization\ndirectly. This is because the in-plane magnetization has opposite signs on the top and\nbottom surfaces so that their sum cancels out. The same argument applies to the\nspontaneous magnetic domains: we are measuring the anomalous supercurrents around\nthe domain boundaries. The magnetic contrast we observe is consistent with those\nexpected from boundary and edge supercurrents.\n5. Finally, what if the magnetic impurities are not intrinsic? They could simply be magnetic\ncontaminants introduced during sample preparation or growth. If that is the case,\ninvestigation of this phenomenon would be meaningless.\n-Any extrinsic magnetic impurities would have generated magnetic signal which has\nno dependence on superfluid density, the Tc or any applied supercurrent. The\ncomprehensive data we present throughout this manuscript is clearly against such\nartefacts, which are straightforwardly excluded from our investigation (see Lines 167-\n170).\nThe magnetic moments in our study originate from the interstitial Fe atoms inherently\nembedded within the Fe(Se,Te) crystal lattice, rather than extrinsic contaminants. This\nintrinsic feature is well established in prior reports on this material system. These\ninterstitial Fe atoms locally break time-reversal symmetry. Crucially, their interaction\nwith the superconducting condensate enables phenomena such as the quantum\nanomalous vortex state. Furthermore, their behavior, particularly on the surface, is\nprofoundly influenced by the interplay with the topological surface bands and strong\nRashba-type spin-orbit coupling. This intricate intrinsic mechanism is the very subject\nof our investigation. To clarify this foundation, we have strengthened the description in\nthe manuscript in Lines 128-138.\nResponse to Reviewer #4\nThe paper entitled, \"Supercurrent-controlled spontaneous ferromagnetism of magnetic\nimpurities in a spin-orbit-coupled superconductor\" describes scanning SQUID\nmeasurements showing evidence for a long-range ferromagnetic order of the impurity\nmoments on the surface of Fe(Se,Te).\nThe presented measurements are of high quality and their interpretation and analysis\nappear appropriate. In particular, the authors do a careful job checking various alternative\nscenarios that may explain their magnetic signal, including vortex trapping. A lengthy\nsupplementary section contains useful information in this respect. Altogether, I find the data\nand analysis convincing and interesting.\nThe scanning probe aspect of the work is expertly done and is a good example of how\nscanning SQUID microscopy can be used to reveal the microscopic mechanisms behind\nsuperconducting phenomena in ways that conventional measurements without spatial\nresolution cannot.\nThe authors have also satisfactorily addressed the detailed comments of reviewer #2, in\nparticular his/her questions relating to the possibility that the observations can be explained\nby vortex entry.\n-We sincerely appreciate the reviewer's positive feedback on our manuscript and the\nrecognition of the quality of our scanning SQUID measurements as well as our\ncomprehensive analysis. We are also grateful for the acknowledgment of our efforts to\naddress alternative scenarios, including vortex trapping, and the utility of our\nsupplementary information. The comments affirm the robustness of our study and the\nvalue of scanning SQUID microscopy in elucidating microscopic mechanisms in\nsuperconducting systems.\nFor these reasons, I recommend that the manuscript be published after the address one\nsmall concern, discussed below.\n1) I did not grasp the difference between the data shown in Fig. 2b from sample #1. Here\nthe sample is measured in the virgin state at 1.68 K. Magnetic domains are observed.\nHowever, in Fig. 3d, after the same zero-field cooling procedure the sample is again in the\nvirgin state, but shows no contrast at all and no magnetic domains. What is the difference\nhere? Why do the samples behave differently? Did I miss something? Could this be made\nclearer in the text?\n-We thank the reviewer for pointing out this potential confusion. For sample #1, the\nweak contrast of magnetic domains is revealed by demodulating the flux signal at the\nvibration frequency of the tuning fork carrying the nano-SQUID, which measures the\nflux gradient along the vibration (z) axis. This demodulated signal is approximately\nproportional to the direct-current (DC) flux but has a much lower noise level,\nallowing domain contrast to be resolved. Meanwhile, the magnetometry data of\nsample #2 were obtained from a direct-current flux measurement through the pickup\nloop. In this configuration, the contrast of magnetic domains is typically below the\nDC noise floor, resulting in the absence of visible domains.\nThis distinction between the two measurement modes is described in the Methods\nsection in Lines 496-503. We have also added an explicit clarification into the revised\nmain text in Lines 160-164.\nBelow please find the point-by-point response to the reviewers’ questions and\ncomments.\nResponse to Reviewer #2\nThe addition of Extended Data Fig. 13 with the homogeneous flake addresses my previous\nconcerns. I can now recommend publication.\n-We sincerely appreciate the reviewer’s constructive feedback throughout the review\nprocess, which has helped us continuously strengthen the manuscript. We are grateful\nfor the recommendation for publication.\nResponse to Reviewer #3\nI have checked the study, and I am not convinced by the reply. Below are some major\nissues:\n1.First of all, it is generally boring, trivial as well as risky to study impurities. Impurities are\ncomplicated and often cause artifacts because there are so many different types of\nimpurities. Over the years, we have witnessed some many false claims due to impurities.\nAnd, impurities are typically boring, because they are not intrinsic and harder to control\nduring the fabrication.\n-We regret that our comprehensive experimental evidence and analysis is not enough\nfor the reviewer. The impurities in our samples are not random or uncontrolled. We\nutilize a well-known type of impurities, the interstitial Fe atoms within the Fe(Se,Te)\ncrystal lattice. The density of these impurities can be effectively controlled during the\ncrystal growth and annealing process.\nImportantly, these impurities break time-reversal symmetry and exert a profound\ninfluence on the superconducting properties. This enables the exploration of novel\nemergent phenomena that cannot be simply attributed to the presence of impurities\nalone (See Refs. 14 and 37-39 for example). Our work advances the understanding of\nferromagnetic coupling mediated by superconducting condensate in topologically\nnon-trivial superconductor with Rashba SOC.\nWe also note that one important type of ferromagnetism is due to magnetic impurities\nin semiconductors induced by RKKY interaction. In fact, our work bears similarity to\nthis type of ferromagnetism in the context of a superconductor.\n2.\nSecond, what kind of magnetic impurities is the author talking about? Did the author\nobserve them in the samples? I do not see experimental evidence. What is the size of the\nimpurities? What is their density? Are they atomic impurities, or clusters of Fe or some\nother elements?\n-The magnetic impurities are interstitial Fe atoms embedded within the Fe(Se,Te)\ncrystal lattice, as described in the manuscript (Lines 128-138). The presence and\ntypical density of these Fe atoms, which can be directly observed by scanning\ntunneling microscopy (STM), are well-established features of this material system as\ndocumented in prior works (Refs. 33, 37-39). In our study, the average distance\nbetween interstitial Fe atoms should be comparable to the superconducting coherence\nlength as discussed in Lines 140-144.\n3.\nThird, even assuming all the assumptions above are correct (which they are not in this\npaper), there is no evidence that the superconductor is spin-orbit coupled or that the\nphenomenon involves spin-orbit coupling. An inhomogeneous current in a superconductor\ncan already produce an inhomogeneous magnetic field and switch magnetization. This has\nbeen used in many effects in many other studies.\n-The existence of Dirac surface states and Rashba spin-orbit coupling (SOC) in\nFe(Se,Te) systems with compositions similar to ours has been previously established\nby ARPES measurements and band-structure calculations (e.g., Refs. 31, 40), and the\ngeneral principles of magnetic coupling with SOC in such systems have been\nanalyzed in Refs. 10-13.\nThe key novelty of our work lies in the direct observation of magnetic domains and\nmagnetization manifested through the supercurrent (which is intrinsically linked to the\nsuperfluid density). This specific manifestation is distinct from phenomena reported\nin prior literature. Crucially, this interplay can tremendously reduce the switching\nthreshold for these domains. This finding positions Fe(Se,Te) as a highly promising\nmaterial platform for the development of energy-efficient cryogenic memory.\n4.\nFourth, again, the authors never patterned their samples or devices to control the current\nflow. With proper patterning, the authors might be able to make a case and systematically\nand quantitatively calculate the current effect. However, in all the samples/devices\npresented, the authors let the current flow randomly. The system is complicated by\nmagnetic impurities, vortices, roughness, and the fact that the authors vary both\ntemperature and current amplitude. Therefore, it is risky to make any major scientific claim\nwithout conducting proper control experiments.\n-The mechanical exfoliation technique, common in the study of 2D materials,\ntypically yields samples with irregular morphologies. However, the key phenomena\nwe report are not contingent upon sample shape regularity. The supercurrent is not\nrandomly flowing but preferentially along the sample edges above a specific\nswitching threshold. This effect is a distinct behavior that we have systematically\ndistinguished from potential alternative explanations such as the influence of vortices\nor sample roughness. As a natural progression for this research line, we note in the\nmanuscript (Lines 419-420) that the fabrication of regular, lithographically-defined\ndevices represents an important next step for future investigation.\n5.\nEssentially, the authors observed an inhomogeneous magnetic-field distribution and some\ncurrent-intensity-dependent magnetic-field distribution. The phenomenon itself is real.\nHowever, there is no clear evidence that the system exhibits ferromagnetism. If the authors\nsimply claimed that they are mapping the supercurrent or its interaction with magnetic\nfeatures, that would likely be correct. But the authors did not interpret it convincingly.\nWithout sufficient proof, calling this “supercurrent-controlled ferromagnetism” is certainly\nan overclaim.\n-Our claim regarding ferromagnetism is not reliant on magnetic morphology alone.\nInstead, it is substantiated by the unique interplay between the magnetic moments and\nthe superconducting condensate, and most critically by their response to experimental\nmanipulation. Specifically, we have demonstrated the existence of zero-field-cooled\nmagnetic domains mediated by the superconducting condensate and the random\nredistribution of these domains over repeated thermal cycles (Fig. 2 and Extended\nData Fig. 1). Furthermore, the key experimental signatures that the controlled\nswitching of magnetic domains, the presence of remanent magnetization, and a clear\nhysteresis loop (Figs. 3 and 4, Extended Data Figs. 12 and 13) are established\nhallmarks of ferromagnetic order.\n5.\nEven if I were to accept the authors’ interpretation, I agree with the other reviewer that\nthe authors have failed to rule out the trivial scenario in which inhomogeneity and vortex\npenetration lead to the observed signal. In their reply to the other reviewer, the authors\nclaimed that remnant magnetization is strong evidence against the trivial scenario of vortex\ntrapping, but I believe the authors are incorrect. As the other reviewer pointed out, the\nactual situation is far more complicated.\n-As detailed in our response to Reviewer #2, we have conducted the suggested control\nexperiments using a homogeneous sample. The results demonstrate that under the\napplied current bias, there is no local suppression of superconductivity. Furthermore,\nwe observe a clear remnant magnetization signal after the current is removed\n(Extended Data Fig. 13). This clearly argues against the vortex trapping scenario.\n6.\nIn some of the results in this study, the observations contradict fundamental properties of\nmagnetism and magnetic fields. In-plane magnetization should produce a dipole-like\nmagnetic-field distribution. This contradicts what is shown in both the extended figures and\nthe main text. In the main-text figures, the features appear in-plane, whereas in the\nextended figures the same features become out-of-plane. The authors claimed that they\nmake an \"intensive\" explanation, but these explanations are based on estimation and\nhypothesis without experimental proof.\n-We note in the manuscript (Lines 298-338) that our data of remnant magnetic\ncontrast after the bias current is removed originates from the anomalous edge\nsupercurrent of a Rashba superconductor, rather than the in-plane magnetization\ndirectly. Similarly, the spontaneous magnetic domains we image are manifested\nthrough the local anomalous supercurrents circulating at domain boundaries, which\nproduce a magnetic signature analogous to an in-plane configuration. When a bias\ncurrent is applied, it aligns the underlying magnetic moments, leading to an overall\naveraged effect that resembles an out-of-plane magnetization—much like the net\ncirculation pattern observed in edge supercurrents. Therefore, our experimental\nobservations are fully consistent with the interpretation."}]

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