# Anisotropic Paramagnetic Peak Effect in Reversible Magnetization of
Crystalline Miassite Superconductor $\text{Rh}_{17}\text{S}_{15}$

Ruslan Prozorov<EMAIL_ADDRESS>Ames National Laboratory, Ames, Iowa
50011, USA  Department of Physics & Astronomy, Iowa State University, Ames,
Iowa 50011, USA  Makariy A. Tanatar  Ames National Laboratory, Ames, Iowa
50011, USA  Department of Physics & Astronomy, Iowa State University, Ames,
Iowa 50011, USA  Marcin Kończykowski  Laboratoire des Solides Irradiés,
CEA/DRF/lRAMIS, École Polytechnique, CNRS, Institut Polytechnique de Paris,
F-91128 Palaiseau, France  Romain Grasset  Laboratoire des Solides Irradiés,
CEA/DRF/lRAMIS, École Polytechnique, CNRS, Institut Polytechnique de Paris,
F-91128 Palaiseau, France  Alexei E. Koshelev  Department of Physics and
Astronomy, University of Notre Dame, Notre Dame, Indiana 46656, USA  Linlin
Wang  Ames National Laboratory, Ames, Iowa 50011, USA  Department of Physics &
Astronomy, Iowa State University, Ames, Iowa 50011, USA  Sergey L. Bud’ko
Ames National Laboratory, Ames, Iowa 50011, USA  Department of Physics &
Astronomy, Iowa State University, Ames, Iowa 50011, USA  Paul C. Canfield
Ames National Laboratory, Ames, Iowa 50011, USA  Department of Physics &
Astronomy, Iowa State University, Ames, Iowa 50011, USA

(1 June 2024)

###### Abstract

We report an unusual anisotropic paramagnetic peak effect observed in
reversible magnetization of a single crystalline nodal superconductor
$\text{Rh}_{17}\text{S}_{15}$. Both temperature- and field-dependent
magnetization measurements reveal a distinct novel vortex state above
approximately 1 T. This peak effect is most pronounced when the magnetic
field, $H$, is applied parallel to the $\left[111\right]$ direction, whereas
it diminishes for $H\parallel\left[110\right]$. Intriguingly, for
$H\parallel\left[100\right]$, instead of a peak, we observe a step-like
decrease in $M(T)$, with the step amplitude increasing in larger applied
magnetic fields. This behavior is opposite to the expectations of conventional
Meissner expulsion. The magnitude of the peak effect, expressed in terms of
dimensionless volume susceptibility, is on the order of $\Delta\chi=10^{-5}$
(with full diamagnetic screening corresponding to $\chi=-1$). The observed
anisotropic paramagnetic vortex response is unusual considering the cubic
symmetry of $\text{Rh}_{17}\text{S}_{15}$. We propose that in this distinct
vortex phase, a small but finite attractive interaction between vortices below
$H_{c2}$ may be responsible for this unusual phenomenon. Furthermore, the
vortices seem to prefer aligning along the $\left[111\right]$ direction,
rotating toward it when the magnetic field is applied in other directions. Our
findings add another item to the list of unusual properties of
$\text{Rh}_{17}\text{S}_{15}$ that attracted recent attention as the first
unconventional superconductor that has a mineral analog, miassite, found in
nature.

##  I Introduction

Since the division of superconductors into two types by Abrikosov in 1957 [1],
vortex physics has become one of the most active research areas in
superconductivity due to its rich fundamental physics aspects and direct
relevance for technological applications [2, 3, 4, 5, 6, 7, 8, 9, 10]. Various
vortex lattice phases and behaviors, usually mapped on the vortex $H-T$ phase
diagram, are believed to be linked to the unconventional nature of
superconductivity in different classes of superconductors, such as
high$-T_{c}$ cuprates [6, 8, 9, 11], borocarbides [12, 13] and iron-pnictides
[14, 15, 16, 17, 11]. One of the most interesting mixed-state features is the
non-monotonic dependence of the irreversible component of magnetization on a
magnetic field or temperature. Depending on the context and author’s
preferences, this feature can be called the “peak effect”, the “second
magnetization peak” or the “fishtail” [18, 5, 19, 20, 21, 6, 7, 22, 23, 24,
25, 26, 27, 28]. Since any measurement has a certain experimental time window,
the measured magnetic moment or current density is affected by magnetic
relaxation, which is exponentially fast at current densities close to the
critical current [6, 7, 8]. Therefore, there is always a question as to
whether the peak effect is due to the actual non-monotonic behavior of
$j_{c}\left(H,T\right)$, which would imply an unusual pinning mechanism, or
whether it is the result of a non-monotonic magnetic relaxation. This latter
“dynamic” mechanism is predicted in the weak collective pinning and creep
model [6] and the former “static” mechanism can be, for example, due to the
softening of the vortex lattice at low fields and close to $H_{c2}$ [2, 5],
two different vortex phases [29] and a crossover from the collective to the
plastic creep mechanism [30, 23]. Importantly, in either case, the reversible
magnetization is always considered to be a monotonic function of a magnetic
field and temperature. In fact, we are not aware of any reports of the non-
monotonic behavior of reversible magnetization, except the so-called
paramagnetic Meissner effect (PME), which seems to be related to extrinsic
factors such as sample inhomogeneities, granularity, demagnetizing factors,
form factor, or inhomogeneous cooling conditions [31, 32, 33].

Superconducting $\text{Rh}_{17}\text{S}_{15}$ with $T_{c}=5.4\>\text{K}$ has
recently attracted attention as the rare case of a cubic compound with line
nodes in its superconducting gap inferred from the $T-$linear variation of the
London penetration depth measured down to 50 mK, and a strong suppression of
$T_{c}$ by non-magnetic disorder [34]. Thermal conductivity measurements down
to 100 mK on the lower $T_{c}=5.0\>\text{K}$ sample show nodal-like concave
field dependence but no residual term [35].

Despite its modest $T_{c}$, $\text{Rh}_{17}\text{S}_{15}$ exhibits quite
unusual superconducting properties. It has a very large upper critical field,
$H_{c2}\left(0\right)\approx 20.5\>\text{T}$, determined from the fit to
Helfand and Werthamer theory [36] of the experimental zero resistivity data
measured along the $\left[111\right]$ direction where the last data point from
the $R(H)$ scan at 0.5 K shows $H_{c2}$ just below 20 T [37]. Data from
polycrystalline samples yielded a similar value of
$H_{c2}\left(0\right)\approx 20\>\text{T}$ [38, 39]. This
$H_{c2}\left(0\right)$ exceeds the Pauli paramagnetic limiting field by a
factor of two, $H_{p}(0)=\Delta(0)/(\sqrt{2}\mu_{B})\approx 10\>\text{T}$,
where the superconducting gap is estimated from the weak-coupling isotopic
Bardeen-Cooper-Schrieffer (BCS) theory [40], $\Delta(0)=1.76k_{B}T_{c}\approx
0.82\>\text{meV}$. There is a large difference between the coherence length of
about $\xi\left(0\right)\approx 4.0$ nm, derived from
$H_{c2}(0)=\phi_{0}/(2\pi\xi^{2})$, and the BCS length scale $\xi_{0}=\hbar
v_{F}/(\pi\Delta(0))$$\approx 21.8\>\text{nm}$, which is more than five times
greater. Here, $v_{F}=0.85\times 10^{5}\>\text{m}/\text{s}$ is the band-
averaged Fermi velocity evaluated by our DFT calculations. In the weak-
coupling BCS theory, in the clean limit (which is the case here), these two
lengths are of the same order, $\xi_{0}/\xi=1.63$ for isotropic $s-$wave [41,
42] and, similarly, with a slightly different numerical prefactor, for
arbitrary $k-$ dependent order parameter, including line nodal $d-$wave [43].
Therefore, an extremely high $H_{c2}(0)$ alone represents a significant
departure from the BCS theory for this relatively low$-T_{c}$ superconductor.
All these properties imply that $\text{Rh}_{17}\text{S}_{15}$ is the first
unconventional superconductor whose formula can be found in nature in the form
of mineral miassite [34].

There is only limited information on the vortex properties of
$\text{Rh}_{17}\text{S}_{15}$, because most studies have been performed on
polycrystalline samples. For single crystals, only critical fields were
reported, but without direction-resolved data. The broad peak effect in
magnetization was observed as a function of the magnetic field and some non-
monotonic signatures were observed in magnetic susceptibility as a function of
temperature [27]. The authors found a strong dependence of the persistent
current density on the time window of the experiment and suggested a mix of
static and dynamic scenarios where the vortex lattice becomes progressively
more disordered with an increasing magnetic field. Perhaps, because of its
cubic crystal structure, it was believed that this does not make much
difference. However, cubic materials can exhibit electronic anisotropy [44],
including in the vortex state, for example of the upper critical field of
niobium [45].

In this paper, we report the significant anisotropy of the vortex response in
single crystals of $\text{Rh}_{17}\text{S}_{15}$ below the upper critical
field, $H_{c2}$. A highly unusual paramagnetic reversible peak effect was
observed in magnetization as a function of temperature and magnetic field for
$H\parallel\left[111\right]$ and it is reduced when
$H\parallel\left[110\right]$. For $H\parallel\left[100\right]$ instead of a
peak, a step in $M(T)$ develops. These obsevations indicate that vortices
prefer to align along the $\left[111\right]$ “easy direction”. The effect is
suppressed by disorder. It appears that there is a novel vortex lattice phase
that has never been reported before.

##  II Results

The basic properties and characterization of our single crystals with a focus
on the superconducting gap structure have previously been reported [34]. In
low fields, there is a sharp superconducting transition in magnetization and
resistivity. In small magnetic fields, there is a significant difference
between zero-field-cooled (ZFC) and field-cooled (FC) magnetization. For
example, following the ZFC protocol by applying $H=10\,\mathrm{Oe}$ after
cooling to a base temperature, $T=2\,\mathrm{K}$, results in volume magnetic
susceptibility, $\chi\approx-1$, indicating a complete diamagnetic screening
in all three orientations. Cooling in the same field from above $T_{c}$ gives
only $\chi\approx-0.018,\,-0.023,\,-0.027$ in the $\left[100\right]$,
$\left[110\right]$ and $\left[111\right]$ directions, respectively. This
indicates that the pinning is weakest in the $\left[111\right]$ direction.
With an increasing magnetic field, the transition curves smear and at
$H=1.5\>\text{T}$ become reversible, showing practically no hysteresis between
ZFC and FC magnetization.

There is a small but notable paramagnetic signal in the normal state, also
reported previously in polycrystalline samples [38, 39]. Just above $T_{c}$,
the volume magnetic susceptibility is on the order of $\chi=1\times 10^{-4}$.
This is comparable to the magnetic susceptibility of tungsten at room
temperature, which is regarded as “non magnetic”. Interestingly, this
paramagnetic signal is temperature dependent. It was suggested that sharp
peaks in the density of states at the Fermi level could result in temperature-
dependent Pauli spin susceptibility. This mechanism has also been suggested
for $\text{V}_{3}\text{Si}$ where the paramagnetic susceptibility per vanadium
ion is of the same order as for rhosium in $\text{Rh}_{17}\text{S}_{15}$ [46,
39]. Perhaps, this active magnetic channel is related to the peak effect
reported here.

In discussing magnetic anisotropy, it is important to analyze the possible
effect of the sample shape, which enters the magnetic response via
demagnetizing factors $N$ [47]. For our fields of interest, above 1 T,
demagnetizing correction can be safely neglected. Specifically, the effective
magnetic field strength, seen by the sample, is
$H_{\text{eff}}=H_{\text{app}}-NM/V$, where $H_{\text{app}}$ is the applied
magnetic field and $V=1.7\times 10^{-9}\>\text{m}^{3}$ is the volume of the
sample. Even if we take an exaggerated value of the measured magnetic moment,
$M=10^{-3}\>\text{erg/G}=10^{-6}\>\text{Am}^{2}$, the corresponding volume
magnetization is $M/V=580\>\text{A/m}$. The applied magnetic field strength
corresponding to a magnetic induction of 1 T is
$H_{app}=B_{app}/\mu_{0}\approx 8\times 10^{5}\>\text{A/m}$. Since the
demagnetizing factor is bound by $0\leq N<1$ (in our case $N\approx 0.35-0.55$
depending on the orientation), the $NM/V$ correction term is of the order of
$200-400\>\text{A/m}$ ($2.5-5.0\>\text{G}$), which is more than three orders
of magnitude smaller than the applied field strength ($10000\>\text{G}$).

Figure 1:  Zero-field-cooled (ZFC) and field-cooled (FC) magnetization of the
same single crystal $\text{Rh}_{17}\text{S}_{15}$ in three different
orientations, (a) $H\parallel\left[100\right]$, (b)
$H\parallel\left[110\right]$, (c) $H\parallel\left[111\right]$. Each curve was
measured in an indicated magnetic field and contains data on warming (ZFC) and
cooling (FC), as shown by red arrows. The ZFC and FC curves coincide within
the noise, indicating practically reversible magnetization. A pronounced
reversible peak effect develops above $H=1.5\>\text{T}$ in $\left[111\right]$
orientation. Red dashed lines show that without the peak, there would be a
continuous transition starting at $H_{c2}$. The peak is reduced in the
$\left[110\right]$ direction. Instead of a peak, $\left[100\right]$ direction
shows a step-like diamagnetic response with the step amplitude increasing with
the increase of an applied magnetic field, unexpected for conventional
Meissner expulsion.

Figure 1 shows the ZFC and FC magnetization of the same
$\text{Rh}_{17}\text{S}_{15}$ single crystal in three different orientations,
(a) $H\parallel\left[100\right]$, (b) $H\parallel\left[110\right]$, and (c)
$H\parallel\left[111\right]$, measured in indicated magnetic fields. Each
curve contains the ZFC and FC data, which are indistinguishable within the
experimental noise, as shown by red arrows. The magnetization is practically
reversible. Note that we chose not to convert magnetization (shown as raw data
in CGS erg/G units) to avoid division by a magnetic field, which would make
the graph less clear.

The central result of this paper is an unexpected pronounced, reversible
paramagnetic peak developing roughly above $H=1.5\>\text{T}$ in a
$\left[111\right]$ orientation, but not present in the $\left[100\right]$
direction where there is a pronounced, steplike diamagnetic decrease in $M(T)$
on cooling, but no peak. In the intermediate orientation, $\left[110\right]$,
there is a reduced peak effect. The noise developing below the transition in
Fig. 1 (a) and (b) is not due to instrumentation but is generated by the
sample and most likely reflects the enhanced motion of vortices. The red
dashed lines in Figure 1(c) show that without the peak, there would be a
continuous transition starting at $H_{c2}$. This indicates that a new distinct
vortex phase is formed inside the superconducting phase, disconnected from the
$H_{c2}(T)$ line.

The peak in a $\left[111\right]$ direction is truly paramagnetic, as it
exceeds the value in the normal state above $T_{c}(H_{c2})$, where the
dimensionless paramagnetic volume susceptibility is of the order of
$\chi=1\times 10^{-4}$, and the peak height from the baseline is approximately
ten times smaller; see the inset in Fig.6. We also stress that, while the
$\left[100\right]$ direction does not show a peak, its temperature dependence
is also unusual. There is a step in $M(T)$ below $H_{c2}$ and, importantly,
its amplitude increases with the increase of the applied magnetic field,
contrary to the expectations of the standard Meissner expulsion. As can be
seen in Fig. 1(a), there is practically no step in $M(T)$ at 1.5 T, but it
grows with the applied field and its amplitude is comparable to the amplitude
of the peak in the $\left[100\right]$ direction.

With such unusual observations, the first step is to verify the effect on
other samples from a different batch. This is shown in Fig. 2 where the
temperature-dependent magnetic moment measured along the $\left[111\right]$
orientation is plotted for two different samples, (a) sample A and (b) sample
B from a different batch. We measured several other samples and confirmed that
the effect is reproducible.

Figure 2:  Zero-field-cooled (ZFC) and field-cooled (FC) magnetization of two
different crystals of $\text{Rh}_{17}\text{S}_{15}$ measured with
$H\parallel\left[111\right]$ in indicated magnetic fields. The curves are
reversible as indicated by red arrows.

The next step is to examine the magnetic field dependence of the magnetic
moment measured at a fixed temperature. Here we observed another surprising
result, the peak effect, but unlike the peak effects and fishtails reported so
far, this peak effect has a significant, dominant, reversible component, which
is consistent with the peak shown in Fig. 1. Figure 3 shows the magnetic
moment measured along the $\left[111\right]$ direction at $T=3.6\>\text{K}$ in
sample A. The external magnetic field was swept as shown in the inset in Fig.
3, repeating down and up sweeps at the same rate, indicated by arrows, and
then switching the ramp rate and repeating the measurement. Four different
ramp rates, from 200 Oe/s, 100 Oe/s, 50 Oe/s and 12 Oe/s, were used, a
seventeen-fold reduction. There is a significant magnetic relaxation with the
blue curve obtained at the slowest 12 Oe/s showing a practically reversible
nonmonotonic $M(H)$ loop.

Figure 3:  Magnetic moment in sample A as a function of the applied magnetic
field measured along the $\left[111\right]$ direction at $T=3.6\>\text{K}$.
The up and down sweeps are performed at the same rate in the directions
indicated by the arrows. Then the sweeps were repeated at different ramp rates
indicated in the inset, which shows the time profile of the magnetic field
sweeps. There is a significant magnetic relaxation with the blue curve
obtained at $12\>\text{Oe/s}$ showing practically reversible $M(H)$ loop.

We now compare the magnetic hysteresis $M\left(H\right)$ loops measured in the
same sample A in three principal orientations. Figure 4(a) shows the
significant anisotropy of the response. There is a conventionally looking
“fishtail” in the $\left[100\right]$ orientation (blue curve), but an
asymmetric, almost reversible, peak effect in the $\left[111\right]$
orientation (red curve). Estimated from the width of the magnetic hysteresis,
the maximum persistent current for the $\left[100\right]$ orientation is about
ten times greater than for the $\left[111\right]$ orientation, and is
intermediate for the $\left[110\right]$ orientation. The location of the peak
shifts to the lower fields from $\left[111\right]$ to $\left[110\right]$ to
$\left[100\right]$. Such significant anisotropy in an electronically isotropic
cubic system is remarkable but apparently not impossible in the vortex state.
In order to better compare the loops, a small paramagnetic background, linear
in $H$, was subtracted. The original loops are shown in the insets with
backgrounds shown by red lines. In the pristine state, a full $M(H)$ loop is
shown, including an often-observed sharp peak near zero field. In this scale,
the background is practically zero. For the irradiated sample, discussed
later, a truncated loop is shown to better visualize the background
contribution. Importantly, the paramagnetic background is practically
identical for all three orientations as well as before and after the electron
irradiation.

Figure 4:  Magnetic hysteresis $M(H)$ loops of the same single crystal of
$\text{Rh}_{17}\text{S}_{15}$ measured in sample A along three principal
directions at $T=3.4\>\text{K}$ with small paramagnetic background subtracted.
The magnetic field ramp rate was 100 Oe/s. Panel (a) shows the results in the
pristine crystal and panel (b) shows the same crystal but after
$2.5\>\text{MeV}$ electron irradiation with the dose of $6.2\times
10^{18}\>\text{electrons}/\text{cm}^{2}$. The $x-$ and $y-$ axes scales are
shared in both panels to allow for direct comparison. Insets show $M(H)$ loops
before background subtraction: (a) a full loop including a pronounced zero-
field peak; (b) a truncated $M(H)$ loop shown to emphasize the background
line.

Another important metric of a superconductor is the response to a controlled
disorder. Previously, we used electron irradiation to probe the
superconducting transition temperature of Rh17S15 and found results consistent
with the line-nodal superconducting gap [34]. Here we examine the effect of
non-magnetic pointlike disorder on the observed paramagnetic reversible peak
effect. Figure 4(b) shows magnetization loops in the peak region for three
orientations after $2.5\>\text{MeV}$ electron irradiation with the dose of
$6.2\times 10^{18}\>\text{electrons}/\text{cm}^{2}$. For direct comparison
with the pristine case, the $x-$ and $y-$ axes scales are shared in both
panels (a) and (b) of Fig. 4, pristine and irradiated, respectively. The
initially almost reversible peak in the $\left[111\right]$ orientation evolves
with disorder into a “normal” fishtail dominated by an irreversible component.
Considering the very low pinning in our $\text{Rh}_{17}\text{S}_{15}$
crystals, it is possible that the peak effect observed in other materials with
larger pinning may hide the reversible phase in their hypothetical clean
state.

The destructive effect of disorder on the observed reversible peak effect is
further illustrated in Fig. 5, where $M(T)$ ZFC-FC temperature scans are
compared for the same sample A in a $\left[111\right]$ orientation in (a)
pristine and (b) electron irradiated states. The peak is smeared and, again,
magnetic noise appears similar to that in Fig. 1(a),(b), probably indicative
of significant vortex displacements and jumps between metastable
configurations.

Figure 5:  ZFC and FC magnetization of the same $\text{Rh}_{17}\text{S}_{15}$
single crystal A measured along $\left[111\right]$ direction, (a) before and
(b) after $2.5\>\text{MeV}$ electron irradiation with the dose of $6.2\times
10^{18}\>\text{electrons}/\text{cm}^{2}$.

Finally, we construct an $H-T$ phase diagram that includes the novel
reversible vortex phase. Figure 6 shows different $H(T)$ lines of
characteristic temperatures defined as shown in the inset. In the inset, the
paramagnetic background above $T_{c}$ was subtracted to show the paramagnetic
volume susceptibility of the peak in absolute units. The blue circles mark the
temperature of the peak maximum, $T_{max}$, and violet circles show the peak
shoulder, $T_{max2}$. The orange area is bound by the onset and offset
temperatures. It represents the domain inside which the novel vortex phase
exists in our range of fields and temperatures. For comparison, additional
lines are obtained tracing the peak position in the $M\left(H\right)$ loops
measured at different temperatures. Green- and violet-filled stars show the
results for the pristine and irradiated samples, respectively. Remarkably, the
peak positions determined from the two measurements coincide in pristine
samples. There is no peak in the $M(T)$ measurements after irradiation and the
peak in the $M(H)$ loops is shifted to lower temperatures. To complete the
phase diagram, Fig. 6 shows the upper critical field in a $\left[111\right]$
orientation before (yellow squares) and after (yellow circles) electron
irradiation. The line $H_{c2}(T)$ is shifted to lower temperatures after
irradiation, consistent with our earlier conclusion that
$\text{Rh}_{17}\text{S}_{15}$ is a line nodal superconductor.

Figure 6:  The $H-T$ state phase diagram including the novel vortex phase. The
definitions of characteristic temperatures of the reversible $M\left(T\right)$
peak are shown in the inset. The blue circles mark the temperature of the peak
maximum $T_{max}$, and the violet circles show the temperature of the knee
(can also be seen as the second maximum) of the peak, $T_{max2}$. Onset and
offset temperatures are the boundaries of the orange area inside which the
peak is located. Green stars and purple stars mark the location of the peak
effect maximum in the $M\left(H\right)$ measurements in pristine and electron-
irradiated samples, respectively. The yellow squares and circles show the
upper critical field in a $\left[111\right]$ direction before and after
electron irradiation, respectively. In the inset, the paramagnetic background
above $T_{c}$ was subtracted to show the volume susceptibility of the peak in
absolute units.

##  III Discussion

Although we do not have a microscopic explanation for the unusual anisotropic
reversible vortex behavior, we can state that we observed a distinct
intermediate vortex state likely related to another order which may be either
magnetic or from a competing superconducting channel. Due to the reversible
nature of the ZFC-FC magnetization and the tendency to such a behavior in
$M(H)$ loops after relaxation of the irreversible component, we can exclude
non-equilibrium flux trapping and vortex density gradients associated with
pinning. This implies that a uniform vortex lattice experiences some
compression, which means that the vortices attract each other. Despite a very
low pinning the field-cooled magnetic susceptibility is surprisingly small,
which may also indicate that the Meissner expulsion is hindered by an
attractive force.

The observed paramagnetic effect is small. In terms of dimensionless volume
magnetic susceptibility, the peak height increases with increasing magnetic
field and appears to saturate around $\Delta\chi\approx+10^{-5}$. This is on
top of the paramagnetic background above $T_{c}$, which is about
$\chi\approx+10^{-4}$. We can estimate how much the distance between the
vortices should change to cause such a peak. For a triangular lattice, the
intervortex distance is, $a=\sqrt{2\phi_{0}/\sqrt{3}B}$. The magnetic
susceptibility for uniform $B$ is, $\chi=\left(B/H_{app}-1\right)$ (as
discussed above, the demagnetizing correction is not important in large fields
of the effect). Therefore, to the first order, the change in the intervortex
spacing, $\Delta a=a-a_{0}\approx-\Delta\chi a_{0}/2$. Positive $\Delta\chi$
means compression. Numerically, for $\mu_{0}H_{app}=9\>\text{T}$, the spacing
is $a_{0}=16.3\>\text{nm}$ and we find that the vortices must move by a small
distance, $\Delta a\approx 9.7\times 10^{-5}\>\text{nm}$. Therefore, if some
attractive force causes the observed paramagnetic peak, it is extremely small
yet finite. An interesting and important aspect is that the situation is
highly anisotropic with a step instead of a peak observed in the
$\left[100\right]$ direction. To explain this, we suggest that the vortices
tend to alight along the $\left[111\right]$ crystallographic direction, so
that when a magnetic field is aligned in other directions and in the absence
of pinning, vortices rotate toward $\left[111\right]$ causing a reduction of
the $\left[100\right]$ component of the magnetization vector. Since our
magnetic measurements are always performed along the direction of an applied
magnetic field, this leads to a step-like feature in $\left[100\right]$
orientation.

Throughout the paper, we emphasized a small but finite paramagnetic
background. Considering the paramagnetic nature of the described effect, it is
possible that normal-state magnetism plays some role, although the mechanism
is unclear.

In the discussion of the unusual effects observed, it is important to estimate
some relevant parameters for $\text{Rh}_{17}\text{S}_{15}$. First, we evaluate
the fundamental length scales. Using the thermodynamic Rutgers relation, we
previously obtained the value of London penetration depth of
$\lambda\left(0\right)=550\>\text{nm}$ [34]. The only other work on single
crystals used magnetization measurements to determine the lower critical
field, which yielded a similar value of $\lambda\left(0\right)=490\>\text{nm}$
[37], although such estimates have a very large error bar. The coherence
length is $\xi(0)=4\>\text{nm}$ for $H_{c2}(0)=20.5\>\text{T}$. Therefore, the
Ginzburg-Landau parameter $\kappa\approx 137.3$ places
$\text{Rh}_{17}\text{S}_{15}$ in an extreme type-II limit. Overall, single
crystals show a very small amount of hysteresis with reversible magnetization
dominating over the irreversible part, even in the regions of the “fishtail”
in $M\left(H\right)$ loops. This is the case for all orientations of the
applied magnetic field. However, in the peak effect region, the hysteresis in
the $\left[111\right]$ orientation is about ten time smaller than in the
$\left[100\right]$ orientation. This clean-limit behavior is also consistent
with the nearly perfect $T-$linear behavior of $\lambda\left(T\right)$ at low
temperatures [34]. Furthermore, from the resistivity at the transition
temperature, $\rho(T_{c})\approx 12\;\mu\Omega\cdot\text{cm}$, the mean free
path can be estimated using the electronic band-structure parameters,
$\ell=3/(v_{F}N\left(0\right)e^{2}\rho)\approx 91\>\text{nm}$, where the band-
averaged product $v_{F}N\left(0\right)=1.1\times
10^{52}\>\text{s}^{-1}\text{m}^{-2}\text{J}^{-1}$. This yields the
dimensionless scattering rate, $\Gamma=0.88\xi_{0}/\ell=0.2<1$, placing the
material in the clean limit.

Vortex-relevant parameters include: the lower critical field,
$H_{c1}\left(0\right)=\phi_{0}\left(\ln\kappa+0.497\right)/(4\pi\lambda^{2})\approx
29\>\text{G}$ for $\lambda(0)=550\>\text{nm}$ and $\xi(0)=4\>\text{nm}$; the
characteristic energy scale,
$\varepsilon_{0}=\phi_{0}^{2}/\left(4\pi\mu_{0}\lambda^{2}(0)\right)=5.6\>\text{meV}/\text{nm}$;
and vortex line energy,
$\varepsilon_{l}=\varepsilon_{0}(\ln\kappa+0.497)=30.2\>\text{meV}/\text{nm}$.
The thermodynamic critical field,
$H_{c}(0)=\phi_{0}/\left(2\sqrt{2}\pi\xi(0)\lambda(0)\right)=0.11\>\text{T}$
and the Ginzburg number that characterizes the width of the critical
fluctuations, $Gi=\left(\gamma
k_{B}T_{c}/\varepsilon_{0}\xi(0)\right)^{2}/8=5.3\times 10^{-5}$; here we
assumed anisotropy parameter $\gamma=1$. A practical formula for this
parameter is: $Gi=3.25\times
10^{-16}[\text{nm}^{-2}\text{K}^{-2}]\left(\kappa\gamma\lambda(0)[\text{nm}]T_{c}[\text{K}]\right)^{2}$.
The largest $Gi$ value is observed in high-$T_{c}$ cuprates, of the order of
$10^{-2}$ or more. For low-$T_{c}$ conventional superconductors it is in the
range of $<10^{-7}$ and much less. For example, in niobium, assuming the clean
case values of $\lambda(0)=33\>\text{nm}$, $\xi(0)=93\>\text{nm}$ and
$T_{c}=9.4\>\text{K}$ [48], $Gi=4\times 10^{-12}$. In iron-based
superconductors, with the typical values of $\lambda(0)=200\>\text{nm}$,
$\xi(0)=3\>\text{nm}$ and $T_{c}=33\>\text{K}$ [49], we obtain $Gi=8\times
10^{-5}$, see also [50]. Therefore, the Ginzburg number of
$\text{Rh}_{17}\text{S}_{15}$ is comparable with iron pnictides and it is
unusually large for low-$T_{c}$ superconductors.

##  IV Conclusions

We report an unusual highly anisotropic reversible vortex phase in single
crystals $\text{Rh}_{17}\text{S}_{15}$. For $H\parallel\left[111\right]$,
there is a pronounced peak in $M(T)$. This peak is reduced for
$H\parallel[110]$. For $H\parallel\left[100\right]$, instead of a peak, a
steep step develops. Both the peak and the step amplitudes increase with the
increase of an applied magnetic field. In all orientations, the warming and
cooling scans are reversible. For $H\parallel\left[111\right]$, there is also
a reversible peak effect (inside the irreversible “fishtail”) in $M(H)$ loops
with the same location as in $M(T)$ scans when plotted on a $T-H$ phase
diagram. Non-magnetic point-like disorder induced by electron irradiation
suppresses the unusual features. We suggest that the observed peak effect may
be caused by a weak attractive interaction between vortices. Furthermore,
$H\parallel\left[111\right]$ appears to be the “easy axis” for vortices. When
a magnetic field is applied in a different direction, the vortices rotate
toward $H\parallel\left[111\right]$, leading to a reduction in the
magnetization projection on the measurement axis, which explains the step
observed in the $H\parallel\left[100\right]$ orientation.

##  V Methods

Single crystal growth: Single crystalline samples of
$\text{Rh}_{17}\text{S}_{15}$ were synthesized out of the Rh-S eutectic by
using a high-temperature solution growth technique. The details of growth and
characterization are provided elsewhere [51, 34]. Briefly, elemental rhodium
powder and sulfur were combined in a fritted Canfield Crucible set[52], sealed
in a silica ampoule, slowly heated (over 12 hours) to 1150 and then slowly
cooled from 1150 to 920 over 50 hours and decanted[53]. Millimeter-sized
single crystals of $\text{Rh}_{17}\text{S}_{15}$ with well-defined facets were
obtained.

Samples: Cuboid-shaped samples with visible $\left(100\right)$,
$\left(110\right)$ and $\left(111\right)$ facets were selected for
measurements. Sample A, used for most figures and quantitative analysis, had
dimensions: $1.35\times 1.25\times 1.10\>\text{mm}^{3}$ along three orthogonal
$\left<100\right>$ directions. It had volume $V=1.72\>\text{mm}^{3}$ and
weighed $12.68\>\text{mg}$. Sample B was $1.35\times 1.0\times
0.95\>\text{mm}^{3}$, $V=1.28\>\text{mm}^{3}$ and weight $9.43\>\text{mg}$. We
measured several other samples for statistics and consistently observed
reported here results.

Magnetization measurements: Magnetic moment was measured using a Quantum
Design vibrating sample magnetometer (VSM) in a $9\>\text{T}$ Physical
Property Measurement System (PPMS). Note that the magnetic moment is measured
in CGS units, $1~{}\text{emu}=1~{}\text{erg/G}=10^{-3}\text{Am}^{2}$. The
samples were glued in proper orientation on Pyrex cylinders fitted inside the
brass half-cylinder, a standar sample holder suitable for our samples. The
orientation was done by eye following well-defined crystallographic planes.
The same sample was repeatedly oriented and measured, then irradiated, and
then the same set of measurements was repeated. Other samples from different
batches were measured.

Electron irradiation: The low-temperature 2.5 MeV electron irradiation was
performed at the “SIRIUS” facility of the Laboratoire des Solides Irradiés
(LSI) at École Polytechnique in Palaiseau, France. The general description of
such experiments and specific details are provided elsewhere [54, 55, 56].
Relativistic electrons are particularly suitable for the creation of point-
like defects in solids, because the energy transfer upon their collisions with
ions matches the knockout energy barriers, typically in the
$10-100\>\text{eV}$ range in metallic compounds. A knocked out ion becomes an
interstitial and, together with the created vacancy, forms the so-called
Frenkel pair. The irradiation is carried out with the sample immersed in
liquid hydrogen at about $22\>\text{K}$ to avoid immediate recombination of
Frenkel pairs and clustering of the produced defects. Upon warming to room
temperature, a metastable population of defects remains after some defects
recombined and migrated to surfaces and extended defects [54, 55]. The degree
of annealing depends on the material and in $\text{Rh}_{17}\text{S}_{15}$ is
about 30%. The amount of the produced disorder is gauged by the increase in
resistivity after irradiation. Public domain software developed by NIST
(https://physics.nist.gov) was used for electron propagation and energy loss
calculations and custom SECTE software developed at the LSI at École
Polytechnique was used for the scattering cross-section calculations. In this
work, a single dose of $6.24\times 10^{18}$ electrons/cm2 was used. The total
cross-section to knock out either Rh or S at 2.5 MeV is 76.4 barn, assuming a
typical displacement threshold of 30 eV. This gives about one defect per
approximately 30 conventional formula units ($Z=2$) with a mean distance
between the defects of approximately 3 nm. The resistivity at $T_{c}$
increased from $12\>\mu\Omega\cdot\text{cm}$ to approximately
$30\>\mu\Omega\cdot\text{cm}$, corresponding to a threefold increase in the
scattering rate $\Gamma$, proportional to the concentration of defects. With
the above estimate, $\Gamma=0.2$ in the clean case, and it increases to
$\Gamma=0.6$ upon irradiation that moves closer to the dirty limit of
$\Gamma>1$. The increase in pinning due to additional disorder is evident in
the magnetization $M(H)$ loops presented.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy (DOE), Office of
Science, Basic Energy Sciences, Materials Science and Engineering Division.
Ames Laboratory is operated for the U.S. DOE by Iowa State University under
contract DE-AC02-07CH11358. Electron irradiation was performed on the SIRIUS
platform supported by the French National network of accelerators for
irradiation and analysis of molecules and materials EMIR&A (FR CNRS 3618).

## References

  * Abrikosov [1957] A. A. Abrikosov, On the Magnetic Properties of Superconductors of the Second Group, Sov. Phys. JETP [ZhETF 32, 1442-1450 (1957)] 5, 1174 (1957).
  * Pippard [1969] A. B. Pippard, A possible mechanism for the peak effect in type II superconductors, Philosophical Magazine 19, 217 (1969).
  * Campbell and Evetts [1972] A. M. Campbell and J. E. Evetts,  _Critical currents in superconductors_ , Monographs on physics (Taylor & Francis Ltd. (London), 1972).
  * Dew-Huges [1974] D. Dew-Huges, Flux pinning mechanisms in type-II superconductors, Phil. Mag. 30, 293 (1974).
  * Larkin and Ovchinnikov [1979] A. I. Larkin and Y. N. Ovchinnikov, Pinning in type II superconductors, Journal of Low Temperature Physics 34, 409 (1979).
  * Blatter  _et al._ [1994] G. Blatter, M. V. Feigel’man, V. B. Geshkenbein, A. I. Larkin, and V. M. Vinokur, Vortices in high-temperature superconductors, Rev. Mod. Phys. 66, 1125 (1994).
  * Brandt [1995] E. H. Brandt, The flux-line lattice in superconductors, Rep. Prog. Phys. 58, 1465 (1995).
  * Yeshurun  _et al._ [1996] Y. Yeshurun, A. P. Malozemoff, and A. Shaulov, Magnetic relaxation in high-temperature superconductors, Reviews of Modern Physics 68, 911 (1996).
  * Hull [2003] J. R. Hull, Applications of high-temperature superconductors in power technology, Rep. Prog. Phys. 66, 1865 (2003).
  * Tinkham [2004] M. Tinkham,  _Introduction to Superconductivity_ , 2nd ed., Dover Books on Physics (Dover Publications, 2004).
  * Kwok  _et al._ [2016] W.-K. Kwok, U. Welp, A. Glatz, A. E. Koshelev, K. J. Kihlstrom, and G. W. Crabtree, Vortices in high-performance high-temperature superconductors, Rep. Prog. Phys. 79, 116501 (2016).
  * Bud’ko and Canfield [2006] S. L. Bud’ko and P. C. Canfield, Magnetism and superconductivity in rare earth-nickel-borocarbides, Comptes Rendus Phys. 7, 56 (2006).
  * Müller and Narozhnyi [2001] K.-H. Müller and V. N. Narozhnyi, Interaction of superconductivity and magnetism in borocarbide superconductors, Rep. Prog. Phys. 64, 943 (2001).
  * Canfield and Bud’ko [2010] P. C. Canfield and S. L. Bud’ko, FeAs-Based Superconductivity: A Case Study of the Effects of Transition Metal Doping on BaFe2As2, Ann. Rev. Cond. Matt. Phys. 1, 27 (2010).
  * Wen and Li [2011] H.-H. Wen and S. Li, Materials and Novel Superconductivity in Iron Pnictide Superconductors, Ann. Rev. Cond. Matt. Phys. 2, 121 (2011).
  * Stewart [2011] G. R. Stewart, Superconductivity in iron compounds, Rev. Mod. Phys. 83, 1589 (2011).
  * Chubukov and Hirschfeld [2015] A. Chubukov and P. J. Hirschfeld, Iron-based superconductors, seven years later, Physics Today 68, 46 (2015).
  * DeSorbo [1964] W. DeSorbo, The Peak Effect in Substitutional and Interstitial Solid Solutions of High-Field Superconductors, Rev. Mod. Phys. 36, 90 (1964).
  * Daeumling  _et al._ [1990] M. Daeumling, J. M. Seuntjens, and D. C. Larbalestier, Oxygen-defect flux pinning, anomalous magnetization and intra-grain granularity in YBa2Cu307-δ, Nature 346, 332 (1990).
  * Krusin-Elbaum  _et al._ [1992] L. Krusin-Elbaum, L. Civale, V. M. Vinokur, and F. Holtzberg, “‘Phase diagram”’ of the vortex-solid phase in Y-Ba-Cu-O crystals: A crossover from single-vortex (1D) to collective (3D) pinning regimes, Phys. Rev. Lett. 69, 2280 (1992).
  * Klein  _et al._ [1994] L. Klein, E. R. Yacoby, Y. Yeshurun, A. Erb, G. Müller-Vogt, V. Breit, and H. Wühl, Peak effect and scaling of irreversible properties in untwinned Y-Ba-Cu-O crystals, Phys. Rev. B 49, 4403 (1994).
  * Tang  _et al._ [1996] C. Tang, X. Ling, S. Bhattacharya, and P. M. Chaikin, Peak effect in superconductors: melting of Larkin domains, Europhysics Letters (EPL) 35, 597 (1996).
  * Giller  _et al._ [1997] D. Giller, A. Shaulov, R. Prozorov, Y. Abulafia, Y. Wolfus, L. Burlachkov, Y. Yeshurun, E. Zeldov, V. M. Vinokur, J. L. Peng, and R. L. Greene, Disorder-induced transition to entangled vortex solid in Nd-Ce-Cu-O crystal, Phys. Rev. Lett. 79, 2542 (1997).
  * Banerjee  _et al._ [2000] S. S. Banerjee, S. Ramakrishnan, A. K. Grover, G. Ravikumar, P. K. Mishra, V. C. Sahni, C. V. Tomy, G. Balakrishnan, D. M. Paul, P. L. Gammel, D. J. Bishop, E. Bucher, M. J. Higgins, and S. Bhattacharya, Peak effect, plateau effect, and fishtail anomaly: The reentrant amorphization of vortex matter in 2H-NbSe2, Phys. Rev. B 62, 11838 (2000).
  * Mikitik and Brandt [2001] G. P. Mikitik and E. H. Brandt, Peak effect, vortex-lattice melting line, and order-disorder transition in conventional and high-T-c superconductors, Phys. Rev. B 64, 184514 (2001).
  * Prozorov  _et al._ [2008] R. Prozorov, N. Ni, M. A. Tanatar, V. G. Kogan, R. T. Gordon, C. Martin, E. C. Blomberg, P. Prommapan, J. Q. Yan, S. L. Bud’ko, and P. C. Canfield, Vortex phase diagram of Ba(Fe0.93Co0.07)2As2 single crystals, Phys. Rev. B 78, 224506 (2008).
  * Naren  _et al._ [2009] H. R. Naren, A. Tamizhavel, S. Ramakrishnan, and A. K. Grover, Vortex phase diagram in weakly pinned $\text{Rh}_{17}\text{S}_{15}$, Journal of Physics: Conference Series 150, 052183 (2009).
  * Gao  _et al._ [2022] L. Gao, T. Ying, Y. Zhao, W. Cao, C. Li, L. Xiong, Q. Wang, C. Pei, J.-Y. Ge, H. Hosono, and Y. Qi, Fishtail effect and the vortex phase diagram of high-entropy alloy superconductor, Appl. Phys. Lett. 120, 092602 (2022).
  * Konczykowski  _et al._ [2000] M. Konczykowski, S. Colson, C. J. van der Beek, M. V. Indenbom, P. H. Kes, and E. Zeldov, Magnetic relaxation in the vicinity of second magnetization peak in BSCCO crystals, Physica C 332, 219 (2000).
  * Abulafia  _et al._ [1996] Y. Abulafia, A. Shaulov, Y. Wolfus, R. Prozorov, L. Burlachkov, Y. Yeshurun, D. Majer, E. Zeldov, H. Wuhl, V. B. Geshkenbein, and V. M. Vinokur, Plastic vortex creep in YBa2Cu3O7-x crystals, Phys. Rev. Lett. 77, 1596 (1996).
  * Koshelev and Larkin [1995] A. E. Koshelev and A. I. Larkin, Paramagnetic moment in field-cooled superconducting plates: Paramagnetic Meissner effect, Phys. Rev. B 52, 13559 (1995).
  * Nusran  _et al._ [2018] N. M. Nusran, K. R. Joshi, K. Cho, M. A. Tanatar, W. R. Meier, S. L. Bud’ko, P. C. Canfield, Y. Liu, T. A. Lograsso, and R. Prozorov, Spatially-resolved study of the Meissner effect in superconductors using NV-centers-in-diamond optical magnetometry, New Journal of Physics 20, 043010 (2018).
  * Koblischka  _et al._ [2023] M. R. Koblischka, L. Půst, C.-S. Chang, T. Hauet, and A. Koblischka-Veneva, The Paramagnetic Meissner Effect (PME) in Metallic Superconductors, Metals 13, 1140 (2023).
  * Kim  _et al._ [2024] H. Kim, M. A. Tanatar, M. Kończykowski, R. Grasset, U. S. Kaluarachchi, S. Teknowijoyo, K. Cho, A. Sapkota, J. M. Wilde, M. J. Krogstad, S. L. Bud’ko, P. M. R. Brydon, P. C. Canfield, and R. Prozorov, Nodal superconductivity in miassite Rh${}_{1}7$S${}_{1}5$, Communications Materials 5, 17 (2024).
  * Nie  _et al._ [2024] J. Y. Nie, C. C. Zhao, C. Q. Xu, B. Li, C. P. Tu, X. Zhang, D. Z. Dai, H. R. Wang, S. Xu, W. Jiao, B. M. Wang, Z. Xu, X. Xu, and S. Y. Li, On the superconducting gap structure of the miassite Rh${}_{1}7$S${}_{1}5$: Nodal or nodeless?, arXiv:2405.08633 (2024).
  * Helfand and Werthamer [1966] E. Helfand and N. R. Werthamer, Temperature and Purity Dependence of the Superconducting Critical Field, Hc2. II, Phys. Rev. 147, 288 (1966).
  * Settai  _et al._ [2010] R. Settai, K. Katayama, H. Muranaka, T. Takeuchi, A. Thamizhavel, I. Sheikin, and Y. Ōnuki, Superconducting properties in Rh${}_{1}7$S${}_{1}5$ under magnetic field and pressure, J. Phys. Chem. Sol. 71, 700 (2010).
  * Naren  _et al._ [2011a] R. H. Naren, A. Tamizhavel, and R. S, Superconductivity in $\text{Rh}_{17}\text{S}_{15}$ and $\text{Pd}_{17}\text{Se}_{15}$: A comparative study, Journal of Physics: Conference Series 273, 012074 (2011a).
  * Naren  _et al._ [2011b] R. H. Naren, A. Thamizhavel, K. A. Nigam, and S. Ramakrishan, Superconductivity in rh17s15and pd17se15: A comparative study, J. Phys. Soc. Jpn. 80, SA014 (2011b).
  * Bardeen  _et al._ [1957] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Microscopic Theory of Superconductivity, Physical Review 106, 162 (1957).
  * Werthamer  _et al._ [1966] N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Temperature and Purity Dependence of the Superconducting Critical Field, $H_{c2}$. III. Electron Spin and Spin-Orbit Effects, Phys. Rev. 147, 295 (1966).
  * Kogan and Prozorov [2012] V. G. Kogan and R. Prozorov, Orbital upper critical field and its anisotropy of clean one- and two-band superconductors, Rep. Prog. Phys. 75, 114502 (2012).
  * Benfatto  _et al._ [2002] L. Benfatto, A. Toschi, S. Caprara, and C. Castellani, Coherence length in superconductors from weak to strong coupling, Phys. Rev. B 66, 054515 (2002).
  * Zarea  _et al._ [2023] M. Zarea, H. Ueki, and J. A. Sauls, Effects of anisotropy and disorder on the superconducting properties of niobium, Frontiers in Physics 11, :1269872 (2023).
  * Butler [1980] W. H. Butler, Upper Critical Field of Nb: Calculated Temperature Dependence and Anisotropy, Phys. Rev. Lett. 44, 1516 (1980).
  * Labbe [1967] J. Labbe, Paramagnetic susceptibility in the $\text{V}_{3}\text{Si}$ type of compounds in the normal state, Phys. Rev. 158, 647 (1967).
  * Prozorov and Kogan [2018] R. Prozorov and V. G. Kogan, Effective demagnetizing factors of diamagnetic samples of various shapes, Phys. Rev. Applied 10, 014030 (2018).
  * Prozorov  _et al._ [2022] R. Prozorov, M. Zarea, and J. A. Sauls, Niobium in the clean limit: An intrinsic type-I superconductor, Phys. Rev. B 106, L180505 (2022).
  * Prozorov and Kogan [2011] R. Prozorov and V. G. Kogan, London penetration depth in iron-based superconductors, Rep. Prog. Phys. 74, 124505 (2011).
  * Koshelev  _et al._ [2019] A. E. Koshelev, K. Willa, R. Willa, M. P. Smylie, J.-K. Bao, D. Y. Chung, M. G. Kanatzidis, W.-K. Kwok, and U. Welp, Melting of vortex lattice in the magnetic superconductor $\text{RbEuFe}_{4}\text{As}_{4}$, Phys. Rev. B 100, 094518 (2019).
  * Canfield and Slade [2023] P. C. Canfield and T. J. Slade, Single Crystal Growth of Synthetic Sulfide- and Phosphide-Based Minerals for Physical Measurements, Minerals 13, 429 (2023).
  * Canfield  _et al._ [2016] P. C. Canfield, T. Kong, U. S. Kaluarachchi, and N. H. Jo, Use of frit-disc crucibles for routine and exploratory solution growth of single crystalline samples, Philos. Mag. 96, 84 (2016).
  * Canfield [2019] P. C. Canfield, New materials physics, Rep. Prog. Phys. 83, 016501 (2019).
  * Damask and Dienes [1963] A. Damask and G. Dienes,  _Point Defects in Metals_ (Gordon and Breach, 1963).
  * Thompson [1969] M. Thompson,  _Defects and Radiation Damage in Metals_, Cambridge Monographs on Physics (Cambridge University Press, 1969).
  * Cho  _et al._ [2018] K. Cho, M. Kończykowski, S. Teknowijoyo, M. A. Tanatar, and R. Prozorov, Using electron irradiation to probe iron-based superconductors, Supercond. Sci. Technol. 31, 064002 (2018).