# Effect of controlled artificial disorder on the magnetic properties of
EuFe2(As1-xPx)2 ferromagnetic superconductor

Sunil Ghimire  Ames 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  Kyuil Cho  Ames
Laboratory, Ames, Iowa 50011, USA  Makariy A. Tanatar  Ames Laboratory, Ames,
Iowa 50011, USA  Department of Physics & Astronomy, Iowa State University,
Ames, Iowa 50011, USA  Daniele Torsello  Politecnico di Torino, Department of
Applied Science and Technology, Torino 10129, Italy  Ivan S. Veshchunov
Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku,
Tokyo 113-8656, Japan  Tsuyoshi Tamegai  Department of Applied Physics, The
University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan  Gianluca Ghigo
Politecnico di Torino, Department of Applied Science and Technology, Torino
10129, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Torino
10125, Italy  Ruslan Prozorov<EMAIL_ADDRESS>Ames Laboratory, Ames,
Iowa 50011, USA  Department of Physics & Astronomy, Iowa State University,
Ames, Iowa 50011, USA

###### Abstract

Static (DC) and dynamic (AC, at 14 MHz and 8 GHz) magnetic susceptibilities of
single crystals of a ferromagnetic superconductor,
$\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$ (x = 0.23), were
measured in pristine state and after different doses of 2.5 MeV electron or
3.5 MeV proton irradiation. The superconducting transition temperature,
$T_{c}(H)$, shows an extraordinarily large decrease. It starts at
$T_{c}(H=0)\approx 24\>\textrm{K}$ in the pristine sample for both AC and DC
measurements, but moves to almost half of that value after moderate
irradiation dose. Remarkably, after the irradiation not only $T_{c}$ moves
significantly below the FM transition, its values differ drastically for
measurements at different frequencies, $\approx 16$ K in AC measurements and
$\approx 12$ K in a DC regime. We attribute such a large difference in $T_{c}$
to the appearance of the spontaneous internal magnetic field below the FM
transition, so that the superconductivity develops directly into the mixed
spontaneous vortex - antivortex state where the onset of diamagnetism is known
to be frequency-dependent. We also examined the response to the applied DC
magnetic fields and studied the annealing of irradiated samples, which almost
completely restores the superconducting transition. Overall, our results
suggest that in $\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$
superconductivity is affected by local-moment ferromagnetism mostly via the
spontaneous internal magnetic fields induced by the FM subsystem. Another
mechanism is revealed upon irradiation where magnetic defects created in
ordered $\text{Eu}^{2+}$ lattice act as efficient pairbreakers leading to a
significant $T_{c}$ reduction upon irradiation compared to other 122
compounds. On the other hand, the exchange interactions seem to be weakly
screened by the superconducting phase leading to a modest increase of $T_{m}$
(less than 1 K) after the irradiation drives $T_{c}$ to below $T_{m}$. Our
results suggest that FM and SC phases coexist microscopically in the same
volume.

##  I Introduction

Coexistence and competition of superconductivity and magnetism is a
fascinating and actively studied topic. It is impossible to give even
remotely-complete reference list, see for example Refs. [1, 2, 3, 4, 5, 6, 7,
8, 9, 10]. Full local-moment ferromagnetism can destroy superconductivity even
when it forms well below the superconducting transition temperature, $T_{c}$,
for example in $\mathrm{ErRh}_{4}\mathrm{B}_{4}$. However, even in this case,
there is a narrow, but rich regime of the microscopic coexistence of two
quantum phases [11, 12, 13, 8, 14]. Itinerant ferromagnetism may also coexist
with superconductivity and such materials exhibit some very unusual properties
[15]. Most of the studied ferromagnetic superconductors are singular
compositions, which somewhat limits the possibility to study the trends and
variations of properties in the continuous phase space, such as temperature
vs. doping, $T\left(x\right)$ [8, 10, 7]. In contrast, there are many
antiferromagnetic (AFM) superconductors where the regime of coexistence is
easier to realize. Superconductivity develops on an AFM background as long as
the internal magnetic field modulation occurs at distances much shorter than
the superconductor’s coherence length, $\xi$, which is often realized in real
materials. In turn, antiferromagnetism is largely unaffected by
superconductivity because screening of the magnetic field is effective on the
length scale of London penetration depth, $\lambda_{L}$ [3, 6]. However, if
the FM state is formed via the RKKY exchange interaction, superconducting
pairing of conduction electrons may also affect the strength of the
ferromagnetic exchange. In general, some form of spin arrangement with a net
ferromagnetic component can be realized in a broad range of compositions in
several families of magnetic superconductors, such as borocarbides [16, 17,
18] and more recently in some iron-based superconductors (IBS), where a decade
of intense studies have clearly shown that magnetism plays an important, if
not pivotal, role in their physics [19, 20, 21, 22, 23, 24, 25, 26]. In the
majority of IBS magnetism arises from the iron sublattice with spins aligned
in the Fe-As plane where superconducting condensate mostly resides. However,
in a few IBS compounds, there is an additional magnetism coming from, for
example, europium as part of their formula [25]. In
$\textrm{EuFe}_{2}\textrm{As}_{2}$, Eu2+ ions ($7\mu_{B}$ full local magnetic
moment) order in an A-type antiferromagnet below 19 K while the iron
sublattice develops a spin-density-wave (SDW) below 190 K [27, 28]. The effect
of $\textrm{Eu}^{2+}$ magnetism is so large that can even be used to detwin
the material by applying an in-plane magnetic field [29]. Thanks to a
possibility of a continuous doping of the parent compound, superconductivity
can be induced in some range of compositions by isovalent substitution of
phosphorus for arsenic. With increasing $x$ in
$\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$, the Eu2+ spins
become canted out of the $ab-$plane producing a net ferromagnetic component
along the $c-$axis. In our crystals with $x=0.23$, in zero applied magnetic
field, superconducting transition occurs upon cooling at
$T_{c}\left(H=0\right)\approx 24\>\text{K}$, followed by the magnetic
transition of europium sublattice at $T_{m}\approx$ 18 K [25]. Although rare,
this is not a singular FM/SC composition in this IBS family. In a related
compound, $\text{RbEuFe}_{4}\text{As}_{4}$, ferromagnetism develops at
$T_{m}\approx 15\>\text{K}$ in a superconducting background with
$T_{c}\left(H=0\right)\approx 36.5\>\text{K}$ [30, 31]. It is worth noting
that while most theories address the coexistence of magnetism and
superconductivity in IBS with respect to the iron ions [22, 32, 26], only few
specifically target magnetism coming from other ions, such as
$\textrm{Eu}^{2+}$ [33, 34, 35].

When studying complex non-stoichiometric materials it is important to be able
to fix the composition and examine the evolution of field and temperature
dependencies when some other non-thermal control parameter is varied. One
obvious example of such parameter is pressure, which has been used intensely
for this purpose. Another is a controlled disorder that provides an important
insight into magnetism and superconductivity [36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47], in particular in IBS [48, 49, 44, 24, 50, 45, 51, 52, 53].
Additional scattering can be induced by various means ranging from chemical
substitution [54] to particle irradiation [55]. Controlled disorder has been
used to study superconductors since the times of the famous Anderson theorem
[37] and Abrikosov-Gor’kov theory [38], where the main attention was paid to
the variation of the superconducting order parameter, hence the experimentally
accessible transition temperature, $T_{c}$ [46, 7, 9, 2, 56, 57, 58]. More
recently, the response to the variation of the scattering rate was studied for
other properties, such as superfluid density and thermal conductivity, which
are directly linked to the superconducting order parameter structure [40, 49,
44, 45, 51, 59, 60, 61]. Due to relative rarity of magnetic superconductors,
there is limited experimental information on the effects of disorder
simultaneously on the superconductivity and magnetism. While we are not aware
of such studies in magnetic borocarbides, in IBS the effect of disorder on
magnetism and superconductivity was studied in several works [62, 63, 64, 65],
but none of them on the ferromagnetic Eu-based 122 compounds, except for a
recent study of the effect of proton irradiation on the subject compound by
some of the authors [66].

Figure 1: Temperature-dependent DC and real part of AC magnetic susceptibility
of pristine EuFe2(As0.77P0.23)2 single crystals measured at very different
frequencies: (a) SQUID magnetometer (MPMS, Quantum Design, DC regime,
$H_{DC}=5\>\mathrm{Oe}$); (b) tunnel-diode resonator (TDR, 14 MHz,
$H_{AC}=20\>\mathrm{mOe}$), and (c) coplanar waveguide resonator (CPWR, 8 GHz,
$H_{AC}\approx 1\>\mathrm{Oe}$). Magnetic susceptibility is normalized to $-1$
at low temperatures for (a) and (b), but is shown in absolute values in a
calibrated experiment in panel (c).

In the present work, we study the effects of electron irradiation on single
crystals of $\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$ and
compare with the proton irradiation performed on similar samples. We find that
in this particular ferromagnetic superconductor, magnetic and superconducting
subsystems coexist almost independently. Superconductivity interacts with the
internal magnetic field produced by the Eu2+ sublattice and the ferromagnetism
is barely screened by the superconducting phase. The non-trivial interaction
is revealed when the artificial point-like disorder enhances both potential
and spin-flip scattering channels affecting $T_{c}$ at a much greater rate
compared to the nonmagnetic IBS. Remarkably, controlled disorder combined with
almost reversible annealing allows examining the properties of both phases in
the regimes of $T_{c}>T_{m}$ and $T_{c}<T_{m}$ in a single composition.

##  II Results

Figure 1 shows the temperature-dependent dynamic susceptibility of pristine
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ single crystals
measured using experimental techniques with very different time windows. Panel
(a) shows the DC results obtained by using Quantum Design magnetic property
measurement system (MPMS) at $H_{DC}=5\>\mathrm{Oe}$, panel (b) shows 14 MHz
tunnel-diode resonator (TDR) data with excitation AC field of
$H_{AC}=20\>\mathrm{mOe}$, and panel (c) shows 8 GHz data at $H_{AC}\approx
1\>\mathrm{Oe}$ obtained by coplanar waveguide resonator (CPWR) technique.

For comparison, the data in panels (a) and (b) were normalized to extrapolate
to $\chi=-1$ at the lowest T, whereas panel (c) shows the calibrated data. All
three susceptibility curves clearly show superconducting transition near
$T_{c}\approx 24\>\mathrm{K}$ and ferromagnetic transition at $T_{m}\approx$18
K. The microwave-frequency CPWR data show extra features and a detailed
analysis of the measurements is given elsewhere [67, 66], while we are
interested in a comparison of the transition temperatures. Below $T_{c}$
diamagnetic susceptibility is rather broad compared to much sharper
transitions of nonmagnetic superconductors. This can be attributed to a
substantial pairbreaking coming from the large-moment paramagnetic background.
In the simple picture, if $\mu\left(T\right)$ is the normal state magnetic
permeability, then the measured magnetic susceptibility is renormalized as
[68, 69]:

$\left(1-N\right)\chi\left(T\right)=\frac{\sqrt{\mu\left(T\right)}\lambda_{L}\left(T\right)}{R}\tanh\frac{\sqrt{\mu\left(T\right)}R}{\lambda_{L}\left(T\right)}-1$
(1)

where $N$ is the effective demagnetizing factor, $R$ is the effective
dimension and $\lambda_{L}\left(T\right)$ is the London penetration depth
without magnetism present. Below $T_{c}$ the
$\sqrt{\mu\left(T\right)}\lambda_{L}\left(T\right)$ term dominates the
behavior with two competing trends. Taking the simplest functional forms, in
the interval $T_{m}<T<T_{c}$,

$\sqrt{\mu\left(T\right)}\lambda_{L}\left(T\right)\sim\left(\left(T-T_{m}\right)\left(T_{c}-T\right)\right)^{-1/2}$
(2)

which is, indeed, a non-monotonic function of temperature in this interval
which is seen in all three measurements shown in Fig. 1. Below $T_{m}$ the
magnetic susceptibility decreases and the overall signal tends to decrease
again. Of course, the physics around ferromagnetic transition is significantly
affected by the proliferation of spontaneous (vortex - antivortex) phases as
was determined in the comprehensive microscopic study [70]. This scenario has
been further explored in [67, 71] Similarly, effects of spontaneous vortex
phase was investigated both experimentally and theoretically in already
mentioned 1144 sibling compound, $\text{RbEuFe}_{4}\text{As}_{4}$ [30, 72, 73,
31].

Figure 2: Normalized magnetic susceptibility of pristine EuFe2(As1-0.23P0.23)2
single crystal from TDR measurements at different DC magnetic fields applied
along the $c-$axis. The inset zooms at the rapid suppression of the peak near
the ferromagnetic transition.

Figure 2 shows TDR measurements of temperature dependent magnetic
susceptibility of pristine
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ at different
magnetic fields applied along the $c-$axis. As expected, $T_{c}\left(H\right)$
decreases with the increasing magnetic field while $T_{m}$ remains practically
unchanged. However, as shown in the inset in Fig. 2, at the same time the
height of the peak near $T_{m}$ decreases and disappears completely above 0.2
T. This is a characteristic behavior associated with a local moment
ferromagnetism as shown previously using TDR technique [74].

###  II.1 Electron irradiation

We now turn to the effects of the artificial disorder induced by the 2.5 MeV
electron irradiation. Details of the experiment are described in section IV.
The irradiation dose is measured during the experiment as a total charge flux
passed through the sample and can be expressed in convenient practical units
of coulomb per cm2 to represent the irradiation dose,
$1\>\textrm{C/cm}^{2}=6.24\times 10^{18}\>\textrm{electrons}/\textrm{cm}^{2}$.

Figure 3 shows partial cross-sections of the defects creation calculated using
SECTE software, developed at Ecole Polytechnique (Palaiseau, France)
specifically for electron irradiation.

Figure 3: Defects creation cross-sections for different ions in
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ as function of
electron energy assuming the displacement energy threshold,
$E_{d}=25\>\mathrm{eV}$. At 2.5 MeV, the partial cross-sections are P: 69.2
barn, Fe: 93.5 barn, As: 112.4 barn and Eu: 190.3 barn. The total cross-
section of defects creation is estimated as 116.4 barn, which leads to the
estimate of the $7.3\times 10^{-4}$ displacements-per-atom (dpa) per 1 C/cm2
of the irradiation.

Of course, the greatest uncertainty is the displacement threshold energy
barrier, $E_{d}$, which varies between 10 and 50 eV for different ions and
compounds [75, 76, 77]. In this work its precise value is not important since
we only need the order of magnitude estimate. We used a typical value of 25 eV
commonly assumed for cross-section calculations for both electron and proton
irradiations [75]. This gives around 0.07 at.% dpa (displacements-per-atom)
per $\mathrm{1\>C/cm}{}^{2}$ of electron irradiation or about 7 defects-
creating collisions per 1000 unit cells (10 atoms in a Z=2 unit cell) and
about twice that value for protons. Therefore the density of the defects is
small and they do not alter chemical composition and do not “dope” the system,
which was proven by Hall effect measurements in another 122 compound, BaK122
[78]. Examination of Fig. 3 shows that irradiation at our energy of 2.5 MeV
produces mostly defects on the Eu sites, whereas lower energy, say 1 MeV,
would produce the least defects on the Eu sites. Such energy-tuneable
irradiation is possible and would lead to ion-specific study of the effects of
disorder.

Figure 4: Magnetic susceptibility of
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ single crystal
after 2.5 MeV electron irradiation with a dose of 3.49 C/cm2 (0.254 at.% dpa)
measured in a DC regime using MPMS (top blue curve) and at 14 MHz using TDR
(red curve). The inset shows the evolution of the TDR peak near the
ferromagnetic transition for different applied magnetic fields.

It is important to note that we studied physically the same crystals before
and after the irradiation, so the observed changes are the results of the
added disorder. Figure 4 shows the temperature dependent susceptibility of
electron irradiated sample with the dose of 3.49 C/cm2 measured using MPMS
(blue curve) and TDR (red curve). Both measurements clearly show a very
significant $T_{c}$ suppression, but only a modest increase of $T_{m}$. This
leads to an outstanding result that the irradiation has driven the
superconducting transition from well above $T_{m}$ to well below. Therefore,
we have a unique situation that both regimes could be studied in the same
sample. One of the important properties is the transition temperature itself
as the function of disorder. While in the regime of $T_{c}>T_{m}$ both
measurements gave similar $T_{c}$, see Fig. 1, we see very a different $T_{c}$
in the irradiated sample measured by the two techniques when $T_{c}<T_{m}$.
Clearly, the difference is due to the dynamic nature of the superconducting
transition. Now the superconductivity develops on a ferromagnetic background,
hence in the presence of a finite internal magnetic field and, therefore, the
nature of the transition is reminiscent of the magnetic irreversibility
temperature, which is known to be very frequency-dependent [79] in materials
with large magnetic relaxation, such as high$-T_{c}$ cuprates [80, 81] and
iron pnictides [82, 83, 84].

Figure 5: Electron irradiation and subsequent annealing studies of the same
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ sample as shown
in Fig.4. Curve (1): pristine state; (2): after 3.49 C/cm2 irradiation; (3)
4.55 C/cm2 total dose where 1.06 C/cm2 was added after step (2); (4) after
annealing at 450 K, and (5) after the second annealing at 523 K. Stars mark
the superconducting transition and the circles mark the ferromagnetic
transition.

In the previous studies, we found that defects introduced by the electron
irradiation can be annealed leading to the recovery towards the pristine
state, sometimes almost completely [85, 86]. Figure 5 shows the evolution of
the dynamic magnetic susceptibility measured using TDR first after two
subsequent irradiation runs and then after two steps of annealing. Curve (1)
shows the pristine state; (2): after 3.49 C/cm2 irradiation; (3) at 4.55 C/cm2
total dose where 1.06 C/cm2 was added after the preceding step; (4) after
annealing at 450 K, and (5) after the second annealing at 523 K. The annealing
was done in argon atmosphere for several hours and then cooling overnight
before opening the chamber. We observe a remarkable practically reversible
transformation from the initial state with $T_{c}>T_{m}$ to the state with
$T_{c}<T_{m}$ and back to $T_{c}>T_{m}$ again. The stars mark the apparent
superconducting transition and the circles mark the ferromagnetic transition.
Therefore, the superconducting state can be switched off by the electron
irradiation and recovered by the annealing, leaving magnetism practically
intact, thanks to the local nature of Eu moments. It is quite different in the
case of itinerant magnetism of iron where the magnetic transition is
suppressed at the same large rate as the superconducting transition [87].
Here, $T_{m}$ slightly increases by less than a degree when magnetism sets in
in the normal metal. This shows that superconductivity weakens (screens) the
exchange interaction suggesting, although indirectly, that two phases coexist
microscopically.

Figure 6: The magnetic-field dependence of TDR dynamic magnetic susceptibility
in a 3.49 C/cm2 electron irradiated sample. The indicated magnetic fields were
applied parallel to c-axis. The inset zooms near the transition region.

When studying superconductors, it is often needed to reveal the behavior of
the normal state “behind” the superconducting response. For example, to
estimate the phonon contribution to the specific heat. A common recipe is to
apply strong enough magnetic field and suppress superconductivity. However, in
ferromagnetic superconductors, with relatively high $T_{m}$ and $T_{c}$, the
specific heat jump at the superconducting transition temperature, is hardly
detected/resolved, since the magnetic contribution to the specific heat can be
large [27, 88]. Therefore, our results provide an alternative method to reveal
the normal state and, if needed, recover back the superconducting state by
annealing. Furthermore, this way of $T_{c}$ suppression by the irradiation
without altering chemical composition can be applied for quantitative specific
heat studies of other ferromagnetic superconductors even with with $T_{c}\leq
T_{m}$, in order to deduce magnetic and superconducting volume fractions by
moving $T_{c}$ further down to show that both FM and SC phases are bulk in
nature (or not). This also allows studying the influence of moderate magnetic
fields on the FM transition that is linked to the character of magnetism [74].
By suppressing the superconducting state by electron irradiation we reveal the
local nature of ferromagnetism in
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$. This follows
from the behavior of the peak in dynamic susceptibility in the normal state.
Figure 6 shows the evolution of a ferromagnetic peak with the applied DC
magnetic field along c-axis. Upon cooling from above $T_{m}$, TDR measurements
in local-moment systems exhibit a sharp peak in zero field. When a small
magnetic field is applied, it reduces the amplitude of the peak as shown in
the inset in, Fig.6. In case of itinerant ferromagnetism there is broad
maximum rapidly smearing and shifting to the lower temperatures [74].

Finally, we compare the upper critical field, $H_{c2}(T)$, in pristine state
(black filled circles in Fig. 7) and after 3.49 C/cm2 electron irradiated
state (blue stars) of the same sample. The data for the pristine samples are
close to the values reported for polycrystalline sample [88]. While it is
expected that $H_{c2}(T)$ may have a step-like feature at $T_{m},$ we did not
have an opportunity to study the $H_{c2}(T)$ line in great detail and it is
impossible to draw any conclusions from our data. Yet, the curve shows an
unusual positive curvature enetring the region of $T\leq T_{m}$, which is not
expected in standard models [89]. We can speculate that the magnetic pair-
breaking scattering is reduced in the long-range ordered phase, because it
requires spin-flip of the scatterer. This will cause an increase of
$H_{c2}(0)$ [47].

Figure 7: Upper critical field, $H_{c2}(T)$, of pristine (black circles) and
3.49 C/cm2 electron irradiated (blue stars)
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ with the magnetic
field applied along the $c-$axis. Note that the slope, $dH_{c2}(T)/dT$ near
$T_{c}$ in irradiated state is larger than the slope in the pristine state.

Furthermore, the slope, $dH_{c2}(T)/dT$ near $T_{c}$ is proportional to
$T_{c}$ multiplied by a function of potential and pair-breaking scattering
[47]. That function increases with the increase of the potential scattering
and decreases with the increase of the pair-breaking one. According to the
Anderson theorem, potential scattering does not change $T_{c}$ whereas pair-
breaking scattering decreases $T_{c}$. Note that in sign-changing $s_{\pm}$
order parameter, the interband potential (non spin-flip) scattering is also
pair-breaking, while the inband potential scattering is not, provided that
each band has no nodes or significant anisotropy. [90]. Figure 7 shows that
the slope at lower $T_{c}$ is actually larger than that in the larger $T_{c}$
pristine state indicating that the pair-breaking scattering increases $H_{c2}$
faster than it suppresses $T_{c}$ adding to the conclusion that electron
irradiation produces a substantial amount of the additional pair-breaking
scattering.

###  II.2 Phase diagram and comparisons with other compounds and irradiation

Indeed, the response to any perturbation, irradiation included, should be
gauged against the results obtained with other types of materials and
irradiations. Here we compare the results with CPWR measurements of proton-
irradiated samples. Protons also introduce largely point like defects and, in
addition, nanometric clusters, which slightly reduce the efficiency of the
overall produced defects acting as scattering centers. Detailed account of the
effects of disorder by doping and proton irradiation in
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ is given
elsewhere [66].

Figure 8: Transition temperatures vs. the estimated atomic concentration of
defects in $\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$ single
crystals obtained by different techniques and different doses of irradiation.
(a) electron irradiation in $x=0.23$ crystal, measurements using (red) tunnel-
diode resonator (TDR) and (blue) DC magnetometry; (b) proton irradiation in
$x=0.23$ (hollow symbols) and $x=0.20$ (full symbols) crystals, measurements
using coplanar waveguide resoantor (CPWR). The way the effective disorder has
been estimated in this case was discussed in [66]. Figure 9: The normalized
rate of $T_{c}$ suppression by electron irradiation (panel (a)) and proton
irradiation (panel (b)). Electron irradiation results from this work are
compared to the data in other compounds, summarized in Ref. [59]. The data for
the important for direct comparison
$\textrm{BaFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$ are from Ref. [63].
The dpa percentages were calculated for each of the listed compounds using
SECTE calculations similar to Fig.3. For CPWR data, hollow symbols were
obtained from a crystal with additional doping-induced disorder (x=0.23). The
way the effective disorder has been estimated and its data merged to that of
the x=0.20 was discussed in [66].

Figure 8 shows the superconducting and ferromagnetic transition temperatures
versus the estimated atomic percentage of induced defects. The displacements-
per-atom (dpa) values are based on the calculated cross-sections using SECTE
for electron irradiation and SRIM for proton irradiation [91]. We therefore
present the data on two panels, since the dpa values are obtained using
different calculations. Regardless of the type of irradiation or experimental
time window, the magnetic transition remains robust and stays around
$T_{m}\approx$ 18 K only slightly increasing in the normal state. The
superconducting transition, while suppressed significantly, decreases at a
similar rate for different experimental methods operating at very different
frequencies as long as $T_{c}>T_{m}$. However, the agreement breaks after the
transition temperatures swap places, $T_{c}<T_{m}$. There are two major
contributions at play here. One is the time window of the measurement, which
leads to lower $T_{c}$ for smaller frequencies (longer relaxation). The second
parameter is the nature of the defects. Considering that electrons and protons
have not only very different masses, but also opposite signs of their charge.
Unfortunately, we do not have ion-type resolved particle cross-sections for
the proton irradiation, but it is quite possible that protons knock out
particular ions at rates different from electrons. Here we can conclude that
the background magnetism affects superconductivity in a way consistent with
the conclusions of the previous studies - the superconducting phase develops
with the magnetic field present, which immediately triggers significant time-
dependencies of all measurable parameters.

Figure 9 compares the normalized rate of the $T_{c}$ suppression, $\Delta
T_{c}/T_{c0}$, plotted versus the estimated density of the induced defects,
which was calculated for each of the listed compounds. Similar to 9, two
panels show the results for electrons and protons, respectively. Panel (a)
summarizes the results of electron irradiation, while panel (b) shows proton
irradiation. We stress that the estimated dpa values are estimated using two
different approaches and, also, do not take into account recombination upon
warming and possible clusterization and agglomeration into larger non-point
like groups. Further controlled studies on similar samples are needed to
compare electron and proton irradiation in a quantitative way. Here we see
that the suppression rate for electron-irradiated
$\textrm{EuFe}_{2}(\textrm{As}_{0.77}\textrm{P}_{0.23})_{2}$ is higher
compared to others, non-magnetic, compounds of the IBS. Most likely this is
because of the formation of magnetic scattering centers on Eu sites, in
addition to the non magnetic channel formed by all defects. Considering
partial cross-sections shown in Fig. 3 this scenario is quite plausible. In
the case of $\textrm{BaFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$
irradiated by 3 MeV protons, we also found that the suppression rate of
$T_{c}$ is larger compared with similar compounds. We attribute this enhanced
suppression of $T_{c}$ to the generation of Fe2P, which is one of possible
magnetic compounds that can also generate magnetic scattering [92].

##  III Conclusions

In summary, we used controlled disorder produced by electron and proton
irradiations to induce two states in the same sample: (1) $T_{c}>T_{m}$ and
(2) $T_{c}<T_{m}$. The ferromagnetic transition, $T_{m}$, is weakly affected
by the irradiation whereas the superconducting transition, $T_{c}$, is rapidly
suppressed. We therefore had a unique opportunity to study the same
ferromagnetism in normal and superconducting background and, vice versa,
superconductivity developing in a paramagnetic or a ferromagnetic background.
The ferromagnetic transition temperature increases by less than a degree in
the normal state compared to when it is born out of superconducting background
signaling of the microscopic coexistence and, perhaps, some weakening of the
exchange interaction by the superconducting phase. Furthermore, we conclude
that in $\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$ local-moment
ferromagnetism of Eu2+ sublattice does not have a direct impact on
superconducting pairing, but it affects the superconducting state via the
spontaneous internal magnetic field that creates Abrikosov vortices and
antivortices in the neighboring domains. When $T_{c}<T_{m}$, the
superconducting transition becomes significantly frequency-dependent
reminiscent of the irreversibility temperature, $T_{irr}(H)$, rather than the
true zero-field transition $T_{c}(H=0)$. It is also possible that the
annihilation of vortex-antivortex pairs at the domain boundaries assisted by
an AC field at $T_{c}<T_{m}$ can further enhance the frequency dependence due
to the dynamic response of shaking-depinned vortex-antivortex lattice, which
is different from that of a conventional mixed vortex state. Another effect of
Eu2+ sublattice is to provide the effective pair-breaking “magnetic” defects
upon particle irradiation. This leads to even faster $T_{c}$ suppression by
disorder than in non-magnetic 122 compounds. This also means that the pairing
state of $\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$ is most
likely $s_{\pm}$ as in other IBS.

##  IV Materials and Methods

Crystal growth, samples. Single crystals of
$\textrm{EuFe}_{2}(\textrm{As}_{1-x}\textrm{P}_{x})_{2}$, x = 0.23, were grown
using self flux method from Eu powder (3N purity), FeAs and FeP precursors,
mixed stoichiometrically with nominal x = 0.25 [93]. The batch was grown
inside stainless steel tube in nitrogen atmosphere with
$T_{max}=1350^{\circ}C$ (heating at a rate of 50 ∘C/hour, keeping at $T_{max}$
for 12 hours) followed by slow cooling at 2 ∘C/hour down to
$T_{min}=1000^{\circ}C$.

Tunnel Diode Resonator(TDR). The real part of the radio-frequency magnetic
susceptibility was measured by using a sensitive tunnel diode resonator (TDR)
[94, 95, 96]. The sample (typically $\sim 0.5\times 0.5\times
0.1\>\mathrm{mm}^{3}$) is mounted on a sapphire rod using a trace amount of
ApiezonTM N-grease in a desired orientation and inserted into the inductor
coil. The coil generates a small AC excitation magnetic field, $H_{ac}\approx
1-10\>\mathrm{A/m}$, the exact value of which depends on the distance between
the coil and a copper tube in which the coil is housed for temperature
stability and electromagnetic shielding. The other end of the sapphire rod is
glued into a copper block containing a CernoxTM thermometer and a resistive
heater.

In the experiment, the resonant frequency of the $LC$ tank circuit with the
sample inside the coil is recorded as function of temperature or external DC
magnetic field, generated by the superconducting magnet outside the cryostat.
The shift of the resonant frequency, $\Delta f=f(H,T)-f_{0}$, from its value
without the sample, $f_{0}$, is proportional to the sample magnetic
susceptibility [94, 97]:

$\Delta f\equiv f(H,T)-f_{0}=-\frac{f_{0}V_{s}}{2V_{c}}\chi(H,T)$ (3)

where $\chi(H,T)=dM/dH$ is the actual magnetic susceptibility of a given
sample with volume magnetization $M=m/V_{s}$, where $m$ is total measured
magnetic moment. In paramagnetic samples $\chi>1$, then the total inductance
of the sample inside the coil increases, and resonant frequency decreases,
whereas in a diamagnetic sample the opposite is true. In a superconducting
sample, the magnetic susceptibility of a superconductor is given approximately
by [94, 97]

$\left(1-N\right)\chi(H,T)\approx\frac{\lambda}{R}\tanh\frac{R}{\lambda}-1$
(4)

where $N$ is the effective demagnetizing factor [98] and $R$ is the effective
dimension calculated numerically for a particular sample geometry [94].
Considering a superconducting sample with magnetic penetration depth
$\lambda\ll R$, where $2R$ is the size of the sample in the direction of
magnetic field penetration (field penetrates from two sides), we obtain for
the penetration depth:

$\Delta\lambda\equiv\lambda(H,T)-\lambda(0,0)\approx
R\frac{2V_{c}\left(1-N\right)}{f_{0}V_{s}}\Delta f=G\Delta f$ (5)

where $G$ is the calibration constant. The main source of uncertainty in $G$
is the approximate factor

$\Delta f_{0}=\frac{f_{0}V_{s}}{2V_{c}\left(1-N\right)}$ (6)

which gives the change in frequency when an ideal diamagnetic sample of the
same shape and volume as the sample under study is inserted at base
(theoretically at zero) temperature into the coil. The approximate Eq.6 is
based on an idealized picture of an infinite solenoid where the sample
perturbs the magnetic flux inside. For a realistic finite coil, Eq.6 is only a
rough approximation. However, $\Delta f_{0}$ can be measured directly by
mechanically pulling the sample out of the coil at the base temperature. Our
cryostat is equipped to do just that, so we determine $\Delta f_{0}$ directly
for each sample. Then,

$G=\frac{R}{\Delta f_{0}}$ (7)

which shows the simplified meaning of constant $G$ as the frequency shift when
magnetic field penetrates the entire sample (of size $2R$, \- from two sides,
travelling distance $R$ from each side).

The measurement was conducted down to 400 mK using Janis wet Helium-3 cryostat
and a DC magnetic field that can be provided by the superconducting magnet
ranges up to 9 T. Further details and applications of TDR technique can be
found elsewhere [94, 95, 96, 97].

Coplanar Waveguide Resoantor (CPWR).The coplanar-waveguide-resonator technique
allows determination of the complex permeability of small samples coupled to
the resonator, within a cavity perturbation approach[99, 100]. The presence of
the sample coupled to the resonator induces changes in the resonance frequency
and quality factor of the CPWR, that are related to the real and imaginary
parts of the total AC susceptibility, respectively[66]:

$\displaystyle\Re\chi$ $\displaystyle\approx 1-\frac{2\Delta
f/f_{0}}{\Gamma_{f}}$ $\displaystyle\Im\chi$
$\displaystyle\approx\frac{\Delta\left(1/Q\right)}{\Gamma_{Q}}$ (8)

where $\Delta f/f_{0}$ and $\Delta(1/Q)$ are the experimental shifts of the
resonance frequency and of the inverse of the quality factor induced by the
presence of the sample under test, and $\Gamma_{f}$ and $\Gamma_{Q}$ are
geometrical factors that can be determined by a self-consistent procedure,
which takes into account also the finite size of the crystal and the
consequent demagnetization effects [101]. The overall real and imaginary parts
of the susceptibility for ferromagnetic superconductors are given by a bulk
magnetic contribution and by a screening given by the superconducting
condensate. The superconducting transition temperature $T_{c}$ corresponds to
the onset of the diamagnetic signal (onset of an increase of the resonance
frequency upon cooling), while the magnetic transition temperature $T_{m}$ is
defined as the onset of a positive contribution to the bulk susceptibility
[67].

Electron irradiation. The 2.5 MeV electron irradiation was performed at the
SIRIUS Pelletron linear accelerator facility operated by the Laboratoire des
Solides Irradies (LSI) at the Ecole Polytechnique in Palaiseau, France. At 2.5
eV electrons are moving with relativistic speed of 0.985$c$ and the total flux
of electrons is about 2.7 $\mu$A of electric current through a 5 mm diameter
diaphragm. The acquired irradiation dose is measured by a calibrated Faraday
trap behind the sample and is conveniently expressed in
$\textrm{C}\cdot\textrm{cm}^{-2}$, where 1
$\textrm{C}\cdot\textrm{cm}{}^{-2}=6.24\times 10^{18}$ electrons/cm2.
Electrons are particularly useful, because unlike heavier particles, they
produce well-separated point like defects, called Frenkel pairs
(vacancy+interstitial). But even with electrons, the irradiation needs to be
conducted at low temperature, in liquid hydrogen in our case to prevent rapid
clusterization of newly formed Frenkel pairs. Upon warming up the
interstitials leave the system via various sinks, such as surfaces, defects,
dislocations etc and a metastable population of vacancies remains. Their
concentration is determined by the highest temperature reached - we re-checked
the irradiated samples after a year on the shelf at room temperature with no
noticeable change. We describe the annealing experiment in the main text. In
122 IBS, we estimate that warming up from 22 K of irradiation run to the room
temperature, about 70% of induced scattering centers survives as determined
from in-situ resitivity measurements [87, 78].

###### Acknowledgements.

We thank A. Koshelev for useful discussions and the EMIR SIRIUS team, O.
Cavani, B. Boizot, V. Metayer, and J. Losco, for running the electron
irradiation. Work in Ames 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. Work in Torino was
partially supported by the Italian Ministry of Education, University and
Research (Project PRIN ‘HIBiSCUS,’ Grant No. 201785KWLE). The 2.5 MeV electron
irradiation was performed at the “SIRIUS” facility in Ecole Polytechnique,
Palaiseau, France, a part of EMIR&A French national network of accelerators
under proposals num. 15-5788,16-4368,18-5155. The 3.5 MeV proton irradiation
was performed at the CN facility of the Legnaro National Laboratories (LNL) of
the Italian National Institute for Nuclear Physics (INFN) in the framework of
the INFN-Politecnico di Torino MESH Research Agreement. Work in Japan was
partially supported by a Grant-in-Aid for Science Research (A) (Grant
No.17H01141) by the Japan Society for the Promotion of Science (JSPS).

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