# Extraordinary physical properties of superconducting YBa1.4Sr0.6Cu3O6Se0.51
in a multiphase ceramic material

V. Grinenko  A. Dudka  S. Nozaki  J. Kilcrease  A. Muto  J. Clarke  T. Hogan
V. Nikoghosyan  I. de Paiva  R. Dulal  S. Teknowijoyo  S. Chahid  A. Gulian

###### Abstract

We report on a novel material obtained by modifying pristine YBCO
superconductor in solid phase synthesis via simultaneous partial substitution
of Ba by Sr and O by Se. The intended stoichiometry was originally
YBa2-xSrxCu3O7-xSex with various values of $x$, among which $x\approx 1$
yielded the best results. Simultaneous application of EDX and EBSD confirmed
that Se atoms indeed enter the crystalline lattice cell. The detailed XRD
analysis further confirmed this conclusion and revealed that the obtained
polycrystalline material contains 5 phases, with the major phase ($>$30%)
being a cuprate YBa1.4Sr0.6Cu3O6Se0.51. Following XRD analysis, we applied a
newly developed approach which enabled us to determine the ionic positions and
occupations in the cuprate phase. The obtained superconductor demonstrates
unique properties, including i) two superconducting transitions with
$T_{c1}\approx$ 35 K (granular surface phase) and $T_{c2}\approx$ 13 K (bulk
granular phase) - this granular phase arrangement naturally yields the
Wohlleben effect; ii) reentrant diamagnetism and resistive state; iii) strong
paramagnetism with Curie-Weiss behavior ($\theta_{CW}\approx$ 4 K) and the
ferromagnetic phase overruled by superconductivity; iv) Schottky anomaly
visible in the heat capacity data and most likely delivered by small clusters
of magnetic moments. Thorough analysis of the heat capacity data reveals a
strong-coupling $d-$wave pairing in its bulk phase (with $2\Delta/T_{c}\approx
5$), and, most importantly, a very unusual anomaly in this cuprate. There are
reasons to associate this anomaly with the quantum criticality observed in
traditional cuprate superconductors at much higher fields (achievable only in
certain laboratories). In our case, the fields leading to quantum criticality
are much weaker ($\sim$7-9 T) thus opening avenues for exploration of the
interplay between superconductivity and pair density waves by the wider
research community.

###### keywords:

doubly-substituted YBCO , heat capacity , XRD structure analysis , Wohlleben
effect , reentrant diamagnetism , quantum phase transition.

††journal: Materials Today

[inst1]organization=Leibniz-Institute IFW-Dresden,addressline=PF 270116,
city=Dresden, postcode=D-01171, country=Germany

[inst2]organization=Shubnikov Institute of Crystallography of Federal
Scientific Research Centre “Crystallography and Photonics”,addressline=Russian
Academy of Sciences, city=Moscow, postcode=119333, country=Russia

[inst3]organization=Nanotechnology Systems Division, Hitachi High-Tech
America, Inc.,addressline=2500 NE Century Boulevard, city=Hillsboro,
postcode=97124, state=OR, country=USA

[inst4]organization=Hitachi High-Tech America, Inc.,addressline=,
city=Clarksburg, postcode=20871, state=MD, country=USA

[inst5]organization=Quantum Design,city=San Diego, postcode=92121, state=CA,
country=USA

[inst6]organization=Physics Research Institute,addressline=National Academy of
Sciences, city=Ashtarak, postcode=0203, country=Armenia

[inst7]organization=Advanced Physics Laboratory, Institute for Quantum
Studies,addressline=Chapman University, city=Burtonsville, postcode=20866,
state=MD, country=USA

##  1 Introduction

With the progress in science and technology the discovery of the phenomenon of
superconductivity was inevitable: sooner or later helium would have been
liquefied, and resistivity of metals been tested at liquid helium
temperatures, which actually has been done by Kamerlingh Onnes in 1911 [1, 2].
The discovery of superconductivity in copper oxide materials was much less
probable since the elemental combination BaxLa5-xCu5O5(3-y) with $x=$ 1 and
0.75, $y>$ 0 cannot be delivered by technical progress, and is, first of all,
the product of extraordinary brainwork and stubbornness of its creators [3].
This work generated more than 20,000 research articles, the most famous of
them related to the Y–Ba–Cu–O superconductor with transition temperature
$T_{c}\approx$ 93 K [4]. Qualitatively, it is a result of compositional
substitute of La by Y, another representative of the same rare earth metals
group in Periodic table of elements, combined with the certain variation of
resultant stoichiometry. The YBCO was a phenomenal success and it became the
most famous superconductor, since it swiftly crossed the liquid nitrogen
barrier at ambient pressure, simplifying greatly the research in novel
superconducting materials and delivering a great puzzle for theoreticians to
decipher its mechanism. This finding stimulated many other cationic
substitutions, which eventually delivered higher transition temperature
superconductivity in compositions Tl–Ca/Ba–Cu–O [5], Hg–Ba–Ca–Cu–O [6], and
Bi–Sr–Ca–Cu–O [7] while eliminating the rare earth element in them.

Anionic substitutions in copper oxides are much less explored experimentally,
though theoreticians have predicted interesting features (see, e.g., [8, 9]).
Meanwhile, oxygen plays a crucial role in the copper oxide superconductivity.
Recognizing importance of oxygen, there have been numerous attempts [10, 11,
12, 13, 14, 15, 16, 17, 18] to partially replace oxygen by other
representatives (namely, S and Se) of the chalcogen group to which O belongs
with contradicting statements. While successful chalcogen substitution is
claimed in Refs. [10, 11, 12, 17, 18], the opposite (no entrance of S into the
lattice) is reported in [13]. Moreover, in another work [15] it is claimed
that Se goes into Y’s position. Thus, in view of this ambiguity, the first
task pursued in our research is to prove that Se indeed enters the crystalline
structure and occupies certain O positions in the lattice. This task is
complicated by the fact that the polycrystalline material under investigation
is heterophase: as the analysis revealed, 5 phases are present. Nevertheless,
by combining simultaneous EDX-EBSD analysis with newly suggested method of XRD
analysis we were lucky to resolve this task with accuracy leaving no doubts on
the validity the structural model. The crystalline lattice parameters, as well
as the positions of Se in the crystalline lattice of superconducting phase,
are identified. Details of our approach are given in Methods. Comparing our
results with the previous reports, one can conclude that the main reason for
success with chalcogen substitution is in simultaneous partial substitution of
Ba by Sr - this kind of double substitution was not exercised in the past. The
second part of our report addresses the unique physical properties of
superconducting phase of YBa1.4Sr0.6Cu3O6Se0.51 based on standard set of
measurements in these multiphase samples. Temperature dependence of
resistance, magnetism and heat capacity reveal two superconducting transitions
with $T_{c1}\approx$ 35 K and $T_{c2}\approx$ 13 K. Above $T_{c1}$, anomalous
paramagnetism is detected with the Curie-Weiss behavior and positive Curie
temperature $\theta_{Curie}\approx$ 4 K. While no ferromagnetism is visible in
DC and AC magnetization, presence of magnetic moments (caused by changing the
Cu-ion ligands from O to Se) can be deduced from these Curie-Weiss data. The
samples demonstrate reentrant diamagnetism (the Wohlleben effect) whose
mechanism can be explained in combination with other measured quantities. The
heat capacity studies allowed us to conclude that the lower $T_{c}$-phase,
which constitutes the majority (volume phase), is of the d-wave nature. Very
interestingly, the electronic heat capacity demonstrates features which may be
associated with the quantum criticality border crossing induced by magnetic
field. Observed fluctuations in the AC magnetic susceptibility further confirm
this conjecture. Magnetic field-induced quantum phase transitions, as well as
quantum criticality in general [18], are a subject of intense research since
they are related with the mechanism of high-temperature superconductivity.
Thermodynamic evidence of possible field-induced border-crossing of quantum
criticality in oxy-chalcogen YBa2-xSrxCu3O7-xSex superconductor constitutes
the third main part of our research. Importantly, it occurs at much lower than
usual magnetic fields [19, 20], and thus opens feasibility avenues for the
research community.

##  2 Results

###  2.1 Composition of samples

Our results are obtained on polycrystalline ceramic samples. Initial
stoichiometries with $x=0$, 0.5, 0.75 and 1 were explored. The most drastic
changes compared to $x=0$ took place in samples with $x=1$, which will be
solely described in this article.

About a dozen pellets were prepared quite reproducibly (the details are
presented in Methods). Their morphology is shown in Fig. 1.

Figure 1: Morphology of freshly cleaved YBa2-xSrxCu3O7-xSex sample (by Hitachi
SU8230 SEM). Inset \- surface pattern (by Hitachi SU3500 SEM) of as-prepared
ceramic sample.

The difference in crystallinity of bulk and surface areas is evident from this
figure. Moreover, the heterophase nature is evident in the bulk of the sample.
The phase content would be one of the topics of performed exploration.
However, the first task is to determine the selenium content in the material.
This is a critical topic since the material was synthesized at high-
temperatures (see Methods), and the ionic radius of Se being higher than O
spreads doubts that it may be emanated from the material. Qualitative
confirmation of Se presence was provided by TEM EDX (Fig. 2).

Figure 2: Abundance of Se and other constituent elements (Y, Ba, Sr, O and Cu)
in thin flakes of the powdered polycrystalline YBa2-xSrxCu3O7-xSex material
with the projected stoichiometry YBaSrCu3O6Se (transmission EDX by Hitachi
HT7700 TEM with Bruker X-flash 6T$|$60 system).

Quantitative EDX exploration requires flat surfaces of the material, which is
not the case for the flakes in Fig. 1. For that purpose special techniques
were used (see Methods) which yielded the results shown in Table 1.

Table 1: EDX data Elements %(at) | O | Cu | Se | Sr | Y | Ba
---|---|---|---|---|---|---
Annealed surface | 50.2 | 23.1 | 0.3 | 9.0 | 9.0 | 8.4
Cleaved surface | 55.5 | 20.6 | 1.0 | 7.0 | 8.6 | 7.3
Ion polished surface | 47.0 | 24.7 | 4.9 | 6.9 | 7.7 | 9.0

The most accurate, last row in Table 1, is encouraging; however, it does not
yet guarantee that Se is hosted in any of the unit cells of crystalline
phases. Two more steps were taken for addressing this topic. The first of them
consisted of simultaneous EBSD and EDX analysis, and the second - in the XRD
exploration.

For the simultaneous EBSD and EDX analysis, the electron beam was positioned
on a crystallite with magnification up to $\times$200,000 (Fig. 3a) so as the
Kikuchi pattern is revealed (Fig. 3b) and the corresponding phase is
identified (Fig. 3c). Without changing the position of the beam the
corresponding EDX spectrum is acquired (Fig. 3d).

Figure 3: Compositional characterization of polycrystalline samples with the
initially intended stoichiometry YBaSrCu3O6Se via simultaneous EBSD/EDX
analysis. (a) Material’s granule at $\times$200,000 magnification with
indication of spot to which the electron beam was focused. (b) Sample
orientation revealed an EBSD Kikuchi patern. (c) EBSD database recognition of
Kikuchi pattern. (d) EDX microanalysis at the same spot.

Since the electron beam focusing spot is about 5 nm, and the phase identified
in Fig. 3c corresponds to the YBa2Cu3O7 structure, one can deduce, in view of
composition shown in Fig. 3d, that most likely Se enters the crystalline cells
of the desired phase YBaSrCu3O6Se. This needs a confirmation.

X-ray diffraction indeed confirms this conclusion and provides very valuable
information about the position of atomic constituents in the crystalline
lattice. The results of XRD analysis are shown in Fig. 4 and Tables 2 and 3
(with more details given in Methods).

Figure 4: X-ray diffractogram of heterophase ceramic material with the
initially intended stoichiometry YBaSrCu3O6Se. All the peculiarities of
experimental profile are explained by reflections generated by 5 phases (see
Table 2 and more details in Methods).

After performing Rietveld refinement, the ratio of phase contribution to X-ray
scattering was identified as 30.6 : 26.0 : 24.9 : 13.7 : 4.8 at.% for phases 1
through 5 shown in Table 2.

Table 2: The comparison between the literature and observed values for the
proposed phases.

Phase | % | Space group, No | Chemical formula | Source | ICSD |  a, Å |  b, Å |  c, Å
---|---|---|---|---|---|---|---|---
1 | 30.6 |  $Pmmm$, 47 | YBa1.4Sr0.6Cu3O6Se0.51 | [21] | 87098 | 3.80174(12) | 3.85009(14) | 11.5739(4)
|  |  |  | Our work |  | 3.84359(4) | 3.83295(6) | 11.47711(18)
2 | 26.0 |  $Pnma$, 62 | BaSeO4 | [22] | - | 8.9461(6) | 5.6911(4) | 7.3313(5)
|  |  |  | Our work |  | 8.8970(2) | 5.6891(2) | 7.30815(18)
3 | 24.9 |  $Pna2_{1}$, 33 | Y2Cu2O5 | [23, 24] | 067119,072058 | 10.8038(6) | 3.49600(18) | 12.4665(7)
|  |  |  | Our work |  | 10.78668(19) | 3.49389(5) | 12.4404(3)
4 | 13.7 |  $Pnma$, 62 | Y2BaCuO5 | [25] | 063425 | 12.1793(7) | 5.6591(5) | 7.1323(4)
|  |  |  | Our work |  | 12.30548(14) | 5.67355(14) | 7.03737(13)
5 | 4.8 |  $Pmn2_{1}$, 31 | BaCu(SeO3)2 | [26] | 202386 | 13.353(2) | 5.247(1) | 8.981(1)
|  |  |  | Our work |  | 13.735(3) | 5.3999(14) | 8.867(2)

Phase 1, YBa1.4Sr0.6Cu3O6Se0.51, is basic for this research. Substantiation of
the choice of one of three most probable models and consecutive analysis of
them are described in Methods. The improvement of structural analysis criteria
(such as $R$-factors, R1/wR, the extrema of difference Fourier synthesis,
$\Delta\rho_{\max}/\Delta\rho_{\min}$) takes place with each next model. Model
1: no Se in any oxygen position [21]. From this, we move to Model 2, where Se
partially replaces Y [15]. Finally, we deal with Model 3, in which Se
partially replaces O. For these modeling steps, we found consecutively
$R1/wR=0.98/0.90\rightarrow 1.09/0.93\rightarrow 0.87/0.73$ and
$\Delta\rho_{\max}/\Delta\rho_{\min}$ = +0.30 / -0.48 $\rightarrow$+0.36 /
-0.54 $\rightarrow$ +0.17 / -0.30 e/Å3. Final model has common criteria for
accuracy based on 20001-point profile expressed by $R_{p}$ = 1.32%, $wR_{p}$ =
1.92%, GOF = 3.14, which confirms high enough reliability of the performed
structural analysis [27]. The relevant crystalline parameters are given in
Table 3.

Atom | Wyckoff position | q | x/a | y/b | z/c | U
---|---|---|---|---|---|---
Y1 | 1h | 1 | 0.5 | 0.5 | 0.5 | 0.02303(6)
Ba1a | 2t | 0.702(1) | 0.5 | 0.5 | 0.183335(4) | 0.022926(8)
Sr1 | 2t | 0.298(1) | 0.5 | 0.5 | 0.183335(4) | 0.022926(8)
Cu1 | 1a | 1 | 0 | 0 | 0 | 0.03563(10)
Cu2 | 2q | 1 | 0 | 0 | 0.348091(19) | 0.03616(7)
O1 | 2q | 1 | 0 | 0 | 0.11896(5) | 0.0281(4)
O2 | 2s | 1 | 0 | 0 | 0.40942(7) | 0.0203(2)
O3 | 2r | 1 | 0 | 0.5 | 0.35568(8) | 0.0346(3)
Se4 | 2k | 0.1628(5) | 0.0391(8) | 0 | 0 | 0.0121(3)
Se5 | 1b | 0.1872(8) | 0.5 | 0 | 0 | 0.0049(4)

  * a

Separate refinement of parameters Ba1 and Sr1 in the full-matrix mode is
possible and stable (the Sr1 atom shifts down along the coordinate z at a
certain increase of its parameter U). In the current case, we preferred
avoiding insertion of additional refinement parameters in favor of keeping the
larger ratio of the measurements to the number of parameters.

Table 3: Wyckoff positions, occupancies (q), atomic coordinates (x/a, y/b,
z/c), and equivalent isotropic displacement parameters (U, Å2) for the basic
atoms in the crystal structure of YBa1.4Sr0.6Cu3O6Se0.51

Possible contribution of other phases is mentioned in Discussion.

###  2.2 Basic superconducting properties.

Figure 5 reveals magnetic and resistive properties of the samples.

Figure 5: Sample properties in AC and DC magnetic fields. (a) Real and (b)
imaginary parts of AC susceptibility (at $f=$3333 Hz and $H_{AC}$=1 Oe) are
shown in various DC magnetic fields from 5 Oe to 50 kOe; inset in (b)
illustrates the transition in greater detail. (c) Peculiar behavior (reentrant
diamagnetism) of DC susceptibility with a strong paramagnetism above
superconducting transition yielding the Curie-Weiss behavior; inset shows
similar transition in a sample without Se. (d) Curie-Weiss behavior of
experimental data on paramagnetism; inset shows linear extrapolation of
experimental data in greater detail (circles correspond to average of 2
measurements). (e) Magnetoresistance in weak and strong magnetic fields. Inset
shows noticeable re-entrant resistivity at $T\lesssim 4$ K (see also [28]).
(f) $\mathbf{M(H)}$ at three characteristic temperatures.

Observations shown in Fig. 5(a-f) are complemented by the heat capacity
measurements, Fig. 6.

Figure 6: Heat capacity of the sample with the initially intended
stoichiometry YBaSrCu3O6Se. (a) Two superconducting transitions (with Tc1 and
Tc2) at $H=0$; insets illustrate details of these transitions in $H=0$, 3 and
9 T fields. (b) Modeling the transition of the major superconducting phase by
the $d-$wave $\alpha-$model at $H=0$. (c) Behavior of electronic specific heat
with phonon contribution subtracted and Schottky anomaly present at bulk
superconducting transition in different magnetic fields (the curve coding is
similar to panel d). (d) The same behavior with the Schottky anomaly
subtracted. Insets in panels (c) and (d) clarify the downturn at $H=9$ T. (e)
Specific heat anomaly with increasing magnetic field (from 0 to 13 T) which
reveals quantum criticality (explained in Discussion).

The data presented in Fig. 5 and Fig. 6 provide very fertile grounds for
discussion.

##  3 Discussion

###  3.1 Diamagnetism

The AC susceptibility $\chi^{\prime}(T)$ (Fig. 5a) clearly indicates
superconducting transition at about 30 K. More sophisticated is the behavior
of the DC susceptibility (Fig. 5c) which reveals a peculiarity known as the
Wohlleben effect (it will be discussed after Magnetoresistivity to better
understand the material). Despite positiveness of DC susceptibility at certain
temperature range, the dependence of magnetic moment vs. applied field, M(H)
(Fig. 5f), has negative derivatives at all temperatures below $T_{c}$ $-$ a
feature which corresponds to the superconducting state.

###  3.2 Magnetoresistivity

The second superconducting phase, which is responsible for the reentrant
diamagnetism in Fig. 5c, is also noticeable in magnetoresistivity (Fig. 5e).
This magnetoresistive behavior has a remarkable low-field feature ($H=0$ Oe
and 100 Oe in Fig. 5e): above a certain temperature $T>T_{0}$, $\rho(T)(H=0)$
and $\rho(T)(H=100)$ match with one another; however, at $T<T_{0}$, the
higher-field curve delays with the transition to $R=0$ state. This is an
unusual feature, caused, most likely, by the electron hopping. At lower
temperatures and higher fields, the hopping through the grain boundary barrier
becomes prohibited (or, alternatively, charge carrier trapping becomes more
effective) yielding this observation. In stronger fields, the anomaly is gone
and the $\rho(T)$ behavior becomes typical, as Fig. 5e demonstrates at $H=90$
kOe. The divergent behavior of $\chi^{\prime\prime}$ (Fig. 5b) is also most
likely related to this mechanism: superconductivity in the intergranular areas
disappears in higher fields, leaving only bulk intragranular effects both in
$\chi^{\prime}$ and $\chi^{\prime\prime}$. Notably, at $H=50$ kOe, the
fluctuations in the AC susceptibility become large and mask off the
superconducting transition.

###  3.3 Paramagnetism and Curie-Weiss behavior

The comparison of major curve in Fig. 5c with its inset indicates strong
paramagnetism. Fitting the experimental data above $T_{c1}$ in
YBa2-xSrxCu3O7-xSex by the Curie-Weiss law $\chi=\chi_{0}+C/(T-\theta_{CW})$,
as shown in Fig. 5d, one can get the Curie-Weiss temperature
$\theta_{CW}\approx 4$ K at parameters $\chi_{0}\approx-3\times 10^{-4}$ and
$C\approx 0.53$. This implies [29, 30]:

$\frac{\mu_{\mathrm{eff}}}{\mu_{B}}\equiv
p_{\mathrm{eff}}=\sqrt{C}\frac{\sqrt{3k_{B}}}{\mu_{B}\sqrt{yN_{A}}}\approx
2.8\sqrt{\frac{C}{y}}\approx 2.8$ (1)

where $y$ is the actual Se content in YBa2-xSrxCu3O7-xSex (which, as follows
from Table 1 and the XRD data, is about $0.5$ $-$ the number used in (1)).
This means that per unit cell the average magnetic moment is
$\mu_{\mathrm{eff}}\approx 1.5\mu_{B}$ (since only $50\%$ of the
superconducting phase possess Se). Also, one can estimate the total angular
momentum $J$ from the expression [30]:

$\sqrt{J(J+1)}=p_{\mathrm{eff}}/g\approx 1.4$ (2)

as $J=1$ (we adopted the electron gyromagnetic ratio $g=2$). Presence of
positive Curie-Weiss temperature typically indicates ferromagnetism at
$T<\theta_{CW}$. However, as follows from Fig. 5a, there is no ferromagnetic
response visible in $\chi^{\prime}$ below $T=4$ K which is likely due to the
competing superconducting order111This fact also indicates that the Curie-
Weiss behavior is a property of the superconducting phase in our heterophase
sample.. However, small ferromagnetic clusters are most likely existent, which
can explain the small upturn (reentrant resistivity) visible at $T\approx 4$ K
in Fig. 5e inset. The upturn is also noticeable in $\chi^{\prime}$ (Fig. 5a)
in small magnetic fields, $H=5$ Oe and $H=100$ Oe. With the field increase, at
$H=0.1$ T, the upturn becomes less steep, and completely disappears in
relatively strong fields ($H=1$ T and $H=5$ T). This can be explained by the
orientation of magnetic moments of clusters, which makes the interaction of
these moments with the external field less effective than for the random
orientation. It points towards a preferable angle of orientation (easy axis)
for the AC-field interaction with the magnetic moments, i.e., towards possible
anisotropy. If this conclusion is correct, then much stronger upturns could
appear at certain orientations of the applied magnetic field relative to the
wave vector of the AC-probe field. Testing of this possible anisotropy could
be considered as an interesting topic, though it is beyond the scope of the
present article.

###  3.4 Wohlleben effect

Numerous explanations have been suggested for understanding the mechanism of
curves similar to ones shown Fig. 5c since the first discovery in granular
high-temperature superconductors [31, 32], and later in many other
superconducting objects including Al [33] and Nb [34] disks, Josephson
junction systems [35, 36], In-Sn spheres [37], etc.

From a theoretical point of view, the magnetic moment (or magnetization
$\mathbf{M=B-H}$) of a superconductor is determined by a superfluid current
$\mathbf{j}_{s}$:

$\mathbf{M}=\frac{1}{2c}\int\left[\mathbf{j}_{s}\times\mathbf{r}\right]dV.$
(3)

Thus, theoretical attempts to explain the effect ought to modify the
traditional, Meissner-type behavior of these currents
$\mathbf{j}_{s}(\mathbf{r},t)$ in a drastic manner to switch from the
diamagnetic to the paramagnetic response. These attempts can principally be
classified into two categories of models, the first of which is based on
currents spontaneously flowing in superconductors in absence of an external
field $\mathbf{H}_{ext}$. This field orients the randomly distributed local
moments $\mathbf{M}_{i}$ so that $\left\langle\mathbf{M}_{i}\right\rangle>0$.
The second category of models depends on more geometrically complex current
patterns (in presence of $\mathbf{H}_{ext}$) than envisioned by the ordinary
theoretical approach. Because of the granular structure (Fig. 1), as well as
the presence of effective magnetic moments in the lattice at low temperatures,
the $\pi-$junction model [38] was a direct candidate to consider in our case.
The likelihood of applicability of this model was further enhanced by the
notion (discussed later in this article) that the $d-$wave superconductivity
fits best to our heat capacity data (Fig. 6b). To judge the applicability of
the $\pi-$junction model, we demolished one of our samples and ground it down
to submicron sizes by milling it in a sapphire mortar. The powder was then
thoroughly mixed with a glue, so as to eliminate intergranular couplings.
Figure 7 compares the resultant DC susceptibility with that of the sample
while still in a ceramic form.

Figure 7: DC susceptibility measurements of pristine and powder-glued sample
in $H\sim$ 30 Oe (similar results were obtained in $H\sim$ 5 Oe).

The depth of the first diamagnetic transition (with an onset at 30-35 K),
normalized to the paramagnetic signal, is drastically reduced. That is, for
the same amount of contributing mass of YBa2-xSrxCu3O7-xSex, the diving
amplitude of the first diamagnetic signal is much smaller (by a factor of 6 to
7). This can be explained by the idea that the first transition is related
with the supercurrents flowing on the surface of the granules as well as the
intergranular trajectories (the latter have been essentially destroyed by
milling and subsequent application of the glue). As a result, the effective
volume able to conduct screening superconducting currents is reduced. At the
same time, the upturn amplitude is smaller only by a factor of two.
Importantly, the temperature of upturn also did not change; most likely, this
is because the second superconducting transition takes place inside of the
granules and is not significantly impacted by the granular size reduction and
the granules separation.

Overall, one can conclude that spontaneous currents due to the
$\pi-$junctions, while being very elegant and possibly applicable to other
systems, are not the cause of the Wohlleben effect in our material. Flux
compression looks more feasible. This model as formulated by Koshelev and
Larkin [39], has been developed in detail by many works (see, e.g., [40] and
Refs. therein). In granular superconductors, complete screening of applied
external field $\mathbf{H}_{ext}$ is not the only possibility, though it
corresponds to the absolute minimum of Gibbs’ energy. In some cases, it may
happen that superconductivity first occurs on the granular surface. Then a
flux may be trapped inside of a granule being confined by the superconducting
shell layer. When the temperature is further reduced, the superconducting
layer grows inwards, and a screening current of the same nature which
initially occurred on the surface layer to screen-off the $\mathbf{H}_{ext}$
rises around the trapped flux. From the topology it is clear that the
direction of this current should be opposite to that of the surface one. In
accordance to (3), this can revert the sign of magnetization, making it
positive.

Using expressions (6) and (13) in the original work [39], one can obtain
maximum paramagnetic to diamagnetic susceptibility ratio $\sim 100\%$ for the
the disk geometry. This value is very close to our data in Fig. 7 for the
pristine sample.

One can expect the higher$-T_{c}$ phase to be on the surface of individual
granules (with the lower Se content), while lower$-T_{c}$ phase (more rich
with Se content) is inside the granules. This structural order is similar to
the In-Sn$-$composition studied by Chu et al. [37]. In their study, the
lower$-T_{c}$ superconducting volume was surrounded by a higher$-T_{c}$ phase,
and the Wohlleben effect was detected during the field cooling. From the point
of view of the underlying physics, this structural order facilitates flux
trapping and subsequent compression. Unsurprisingly, the resultant curves
(Fig. 2 of Ref. [37]) resemble ours shown in Fig. 5c and Fig. 7.

###  3.5 Heat capacity anomaly

The heat capacity measurements and related results (Fig. 6) yield the major
implications of our work, as will be clear from further discussions. The low-
temperature superconducting phase, corresponding to $T_{c2}=12.8$ K,
constitutes the majority. Thus, in our analysis of the heat capacity data, we
will focus on this phase. Excluding the phonon and Schottky contributions (the
details are provided in Methods), we find (Fig. 6b) that a rather good
theoretical fit to our experimental data is possible within the $d-$wave
$\alpha-$model (see, e.g., [41, 42]). As follows from this fit, the
superconducting phase content is about $42(wt.\%)$ of the normal phase. This
is not very far from the XRD results (30.6 at.% in Table II correspond to 44.4
wt.%.

Figure 6c,d contains a very unusual result. Namely, in the range $H=0-5$ T,
with the $\textbf{H}-$field increase, the transition temperature $T_{c2}$
moves towards lower values and the ratio $C_{el}(T)/\gamma T$ equals $1$ at
$T>T_{c2}$ in accordance with expectations. However, for further increase of
magnetic field (i.e., at $H=7$ T and much more pronouncedly at $H=9$ T),
$C_{el}(T)/\gamma T$ exhibits a downturn at temperature $T>T_{c2}$ (the
reproducibility of this result has been confirmed on more than one sample;
moreover, to eliminate possibility of data processing artifacts, we checked
visibility of the downturn by comparing the 9 T $C_{p}/T$ raw data with those
in smaller fields: 7 T and below). The field 9T is still below $H_{c2}$, as
follows from the resistivity curve in Fig. 5e, $T_{c2}\sim 6$ K at $H=90,000$
Oe. At this temperature, the electronic heat capacity has a local maximum,
Fig. 6c, which is still noticeable after the subtraction of the Schottky part
(Fig. 6d). Remarkably, the subtraction of Schottky contribution does not
significantly affect the onset temperature of the downturn. In absence of the
downturn, the curve in case of $H=9$ T (as well as $H=7$ T) would have had the
traditional shape (i.e., similar to those at smaller values of $H$).

Thus, we have an anomaly, which is shown in Fig. 6c,d, and is magnified in
insets to them. Because of importance of this downturn (see the next
subsection) we also performed measurements at higher fields $H=10$ T, 11 T, 12
T and 13 T. The results are summarized in Fig. 6e for more convincive evidence
of the effect to be discussed in the next subsection.

Prior to proceeding to it, we will mention that the higher $T_{c}$ transition,
the $T_{c1}$-phase, is also visible in the top inset to Fig. 6a. As was
mentioned at discussing the Wohlleben effect, it may be just the crust
contribution of the same bulk Phase 1 with less content of Se. Its field
dependence contains another anomaly: the higher transition temperatures at
higher fields. However, further discussion of this minor phase behavior is
beyond the goals of this report.

###  3.6 Quantum criticality

Magnetic field-induced quantum phase transitions have been reported in heavy-
fermion [43, 44, 45], iron-based [46, 47], and cuprate [48, 49, 50, 51]
superconductors, as well as in metamagnets [52, 53], antiferromagnets [54,
55], and other systems [56]. Exploration of this important quantum-mechanical
feature of superconducting state appears instrumental for deciphering the
mechanism of high-temperature superconductivity.

In the case of quantum criticality, nearly degenerate ground states exist not
only at very low (ideally zero) temperatures, but reveal themselves over a
range of temperatures. Variation of external or internal parameters may cause
border-crossing between the phases on phase diagrams. In the case of the
second order phase transition, the system may possess a set of quantum
critical points (QCPs) at certain values of tuning parameters [50]. It was
recently recognized that the pseudogap (PG) phase of cuprates ends at QCP
[20]. Moreover, the associated fluctuations can be involved in $d-$wave
pairing and the anomalous scattering of charge carriers. Heat capacity
measurements in the magnetic field [20] are instrumental for the observation
of features relevant to quantum criticality. However, for cuprates, the
critical fields are high, and the required values (e.g., 18 T in case of Eu-
LSCO and Nd-LSCO [20], and 45 T in case of Tl-2201 [19]) exceed typically
available limits of common laboratory apparatus. In case of Tl-2201, the
heavily doped composition is chosen to reduce the value of $T_{c}$ (and
associated destructive magnetic field) from maximally available $T_{c}=93$ K
down to 15 K. In cuprates, compositional modifications may serve as an
alternative to oxygen overdoping, thus reducing the values of required fields
while keeping the typical features observable.

To understand the relationship of the heat capacity anomaly of the bulk phase
with the quantum criticality, one should take into account that
$C_{el}(H)/T\equiv\gamma(H)\varpropto n(H)$, where $n$ is the charge carrier
density. If at $H>H_{0}$ (where $H_{0}\simeq 7$ T) $n(T)$ decreases with the
temperature then a non-Fermi liquid behavior takes place: in Fermi liquids,
$n$ (and $\gamma$) does not depend on temperature. This kind of effect caused
by application of strong magnetic fields was reported and explained by the
fluctuating pair field and PG, which causes the downturn via the border
crossing due to the doping [57, 58, 59]. In our case, the crossing of the
border is due to the cooling in a high magnetic field. We suggest the
explanation of its mechanism which is related with the existence of pair
density wave (PDW) [60], and the charge density waves (CDW) as its consequence
[61]. The emergence of CDW requires the doping range $0.08<$ $p_{CDW}<0.16$.
Within this doping level, in high-enough magnetic fields (i.e., at $H>H_{0}$),
the CDW border crossing occurs as soon as the temperature drops below a
certain value. In pure YBCO, this occurs at $T<20$ K in $H=50$ T [62, 63].
During this process, the re-arrangement of phases inherent to cuprate
superconductors relative to the QCPs on the field-doping phase diagram takes
place.

Appearance of CDW reduces the density of charge carriers. This is associated
with the creation of electron pockets222Interestingly, in addition to the
electron pockets on the Fermi arcs, experimental evidence is obtained in favor
of two nearby hole-type pockets [64], so that the electronic specific heat is
a combination of three contributions, of which the hole pockets contribute
about $17.3\%$ each to $65.4\%$ of the electron pocket contribution. Unlike
the Hall effect, these electron and hole pocket charge carriers contribute
constructively to the specific heat thus increasing the overall effect of the
magnetic field. on Fermi arcs, and this mechanism reveals itself in Hall [62,
63], Nernst [64] and Seebeck [65, 64] effects. In particular, the Hall
coefficient $R_{H}$ for $H$ $>H_{0}$, decreases from its zero value and
becomes negative at low-enough temperatures. The increase of the absolute
value of $R_{H}\propto 1/n$ corresponds to the decrease in charge carrier
density $n$ (see [62]).

An alternative explanation involves the incommensurate SDW [66] rather than
CDW. The latter will be prohibited if $p<0.08$. If Se, like O, provides doping
to the CuO conductivity layer, the doping parameter $p$ (i.e., the number of
holes per Cu) may be estimated by the universal trend reported in Ref. [67]:

$\frac{T_{c}}{T_{c}^{\max}}=1-82.6(0.16-p)^{2}.$ (4)

Here, $T_{c}^{\max}$ is the value of $T_{c}$ at the optimal doping for our
material. The CDW condition $0.08<p<0.16$ [68, 69] infers a range of possible
values for $T_{c}^{\max}$: $12.8$ K $<T_{c}^{\max}<27.2$ K. The inhomogeneity-
caused broad range of $T_{c}$ values visible in Fig. 6b allows the emergence
of quantum criticality at 15 K which is in accordance with the downturn
temperature $\sim$15 K of 9 T and higher field-caused QCP in Fig. 6c-e. As
known for HTSC cuprates [57, 58, 59], at $T=0$, $p_{PG}=0.19$. Corresponding
value of $T_{c}^{max}$ from (4) is then $\sim$14 K, which, within the
experimental inaccuracy, is close to the declared 15 K. The fact that the QCP
observational temperature is dramatically higher than $T=0$ in both competing
cases is not surprising in view of the well-established “fanning out” effect
[70] reported previously [50, 62] in HTSC cuprates.

For pure single-crystal measurements, involvement of quantum criticalities
yields peculiarities in resistivity $\rho$. For example, a drastic upturn
takes place at PG when superconductivity is fully suppressed [58] by
$H>H_{c2}$: the upturn starts gradually much below the pseudogap opening
temperature $T_{\rho}$. Also, two competing mechanisms (reduction of inelastic
scattering and reduction of charge carrier density) may coexist, so the
resultant behavior of $\rho(T)$ may be both upturn and downturn. In our case,
$H_{max}=9$ T $<H_{c2}$. Additionally, in our case, we have the second,
higher-$T_{c}$ phase which can mask-off any gradual changes in $\rho$;
moreover, as follows from the inset in Fig. 5e, granular crusts are causing
semiconductor-type temperature behavior of $\rho(T)$ above $T_{c1}$ thus
obscuring involvement of arguments related with the reported behavior of
resistivity. Having single crystals of this composition where the value of $p$
depends on Se concentration $y$ would further facilitate research in these
cuprates. In particular, one can explore oscillations of the Seebeck and
Nernst coefficient (see Fig 2 in [64]) and other effects caused by quantum
criticality in much more accessible magnetic fields. In the course of our
research the doping parameter was kept constant; principally, additional
opportunities may come from variation of Se concentration as a QCP tuning
parameter.

In general, simultaneous doping of mother substance YBa2Cu3O7 by Sr and Se
provides more flexibility in modifying this remarkable system than mono-
element substitution. The price to pay for this flexibility is multiphase
nature of the sample. However, after this study, one can try to target more
precisely the composition YBa1.4Sr0.6Cu3O6Se0.51 with the adjusted initial
stoichiometry - a fair task for future experimental research. Also, explaining
why its properties are drastically different from the isomorphic YBa2Cu3O6.51
would be very challenging for the theory.

##  4 Conclusions

In summary, we obtained striking anomalies delivered by simultaneous double
substitution of Ba by Sr and O by Se in a pristine YBa2Cu3O7 superconductor.
Simultaneous application of EDX and EBSD proved that Se enters into the
crystal lattice of the superconducting phase of polycrystalline heterogeneous
material. Two superconducting transitions are revealed. One phase, with
$T_{c1}\approx 35$ K, and the second one, with lower $T_{c2}\approx 13$ K. The
latter phase, YBa1.4Sr0.6Cu3O6Se0.51 constitutes the majority of the
superconductor. The second transition, most likely is caused by the granular
surface of the same substance with less amount of Se on the surface of
granules. Heat capacity explorations proved the $d-$wave nature of the main
phase, as well as the existence of electronic Schottky anomaly in it.
Susceptibility measurements revealed magnetic clusters below the Curie
temperature $\approx 4$ K. These magnetic moments, related with Se ligands of
Cu, are reported for the first time. In view of provided XRD information on
crystallographic positions of elements in superconducting phase, fruitful
theoretical exploration should become possible, as it happened with As
vacancies in LaO0.9F0.1FeAs1-δ ($\delta\approx 0.06$) [30]. The most important
finding in our report is related with the anomaly in the heat capacity data in
moderately strong ($7-9$ T) magnetic fields. Various scenarios related with
the competing PDW and PG QC border crossing can explain the obtained effect.
Further experimental work and theoretical analysis on coexistence of PDW/PG
and superconductivity will be easier in YBa2-xSrxCu3O7-xSex than in pure YBCO
due to the less stringent magnetic field requirement, preferably on single-
crystalline or single-phase samples, and could reveal more understanding of
interrelationship between the physical effects reported here and the mechanism
of superconductivity in cuprates.

##  5 Methods

###  5.1 Materials preparation

Polycrystalline samples were prepared via standard solid-state synthesis
routes. Powders of Y2O3, SrCO3, BaCO3, CuO and SrSe were mixed in
stoichiometric proportions to form, after calcination, initial composition
YBa2-xSrxCu3O7-xSex ($x=0$, 0.5, 0.75, 1). After thorough mixing (hand mixing
in sapphire mortar for $40-50$ min per $400$ mg of total constituents, or,
alternatively, combination of initial $10$ min of hand mixing with $4\times 5$
min of mechanical mill mixing (in plastic vials with agate balls), the mixture
was calcined at $900^{0}$C for $100$ min in KSL-1100 furnace in air, then re-
ground, milled again, and pelletized for the second thermal treatment, which
was done at $950^{0}$C for $30$ min. It was then continued at $650^{0}$C for
80 min. Though the elemental composition of synthesized samples may be
different from the initial targeted stoichiometry, we still call the resultant
composition as YBa2-xSrxCu3O7-xSex. Several pellets were prepared, with
diameter $4$ mm, height $1.5$ mm for resistivity and magnetic measurements,
and with diameter $1.5$ mm and height $2$ mm for heat capacity measurements.
The most interesting features were discovered in case of the initial
stoichiometry with $x=1$, so only this case is being discussed in this report.

###  5.2 Compositional Analysis

For quantitative EDX analysis, ion milling was performed. The sample surface
was polished so as to avoid the undesired self-absorption which affects proper
outcome for light elements (O and Se in particular). For this task, the fine-
ground powder was suspended in isopropyl alcohol, and dropped into a gap
between a stainless steel holder and aluminum foil touching it as shown in
Fig. 8a.

Figure 8: Polishing the powder via ion-milling. (a) Arrangement for cutting by
ion knife. (b) Surface of powder prepared for EDX after the ion cutting.

The cutting by ion knife was arranged in Hitachi Broad Ion Milling System,
IM4000Plus, which cut the structure horizontally, providing a polished surface
for microanalysis (Fig. 8b). The SEM EDX data were acquired from a large
surface area, and the revealed composition closely resembled the initial
chosen stoichiometry of the specimen: the data reflected in Table 1 (last
line) confirm that a rather significant amount of Se remains in the material.

A Hitachi SU5000 variable pressure field emission scanning electron microscope
(VP-FESEM) with Oxford Instruments EDX and EBSD under AZtec software was used
for simultaneous EDX/EBSD analysis with electron beam focal diameter $\sim
1-10$ nm which is suitable for the proposed task since it is comparable with
the size of the lattice cell. During this stage of exploration, no ion
polishing was applied. Instead, in order to avoid self-absorption, we changed
the incidence angle of the electron beam until a Kikuchi pattern appeared
(Fig. 3b) for identifying the crystalline structure (Fig. 3c). We then
determined the chemical composition via EDX (Fig. 3d) at the same region of
interest.

###  5.3 X-ray diffraction studies

Diffraction experiments were performed with polycristalline samples in
accordance with the data given in Table 4.

Table 4: Parameters of the powder diffraction experiment Diffractometer | Rigaku MiniFlex 600
---|---
Radiation | Cu-Kα
Wavelength, Å | 1.540593
Detector name | D/teX Ultra2
Detector type | point
Scan range, deg | 10.0 - 110.0
Scan step, deg | 0.005
Scan speed, deg/ min | 1.0
Incident slit, type, deg | Soller slit, 5.0
Receiving slit, type, deg | Soller slit, 5.0

Qualitative phase analysis was performed with the code HighScore Plus 3.0e
[71]. Heterophase nature of the sample was revealed and for the explanation of
all detected peaks, five phases have been involved.

Refinement of structural models of these phases was performed using Jana2006
software [72]. Complex heterophase content of sample yielded strong
correlations between the refining parameters of the models. For the reduction
of the influence of these correlations on the results of our analysis we used
various methods, the main of which was cyclic combination of Rietveld’s
approach [73, 74] and Le Bail’s approach [75] which significantly increased
the stability of the refinement and much improved the fit. The complexity of
the problem arose from the small amount of Se dopant which cannot provide
strong response at the scattering of X-rays. Nevertheless, because of the
achieved high level of relative precision, we were able to detect the
difference between the three cases shown in Table 5.

Table 5: Refinement comparison for three models of Se location in the
structure YBa1.4Sr0.6Cu3O6Se0.51

No | Model |  R1, % |  w, % |  $\mathbf{\Delta\rho_{\mathrm{max}}}$, e/Å3 |  $\mathbf{\Delta\rho_{\mathrm{min}}}$, e/Å3
---|---|---|---|---|---
1 [21] | Additional O positions: O4($x$,0.5,0) and O5(0.5,0,0); Se is absent in the structurea | 0.98 | 0.90 | +0.30 | -0.48
2 [15] | Se partially replaces Yb | 1.09 | 0.93 | +0.36 | -0.54
3 Our model | Se is localized at positions Se4($x$,0.5,0) and Se5(0.5,0,0)c | 0.87 | 0.73 | +0.17 | -0.30

  * a

At full matrix refinement, because of correlational shift, the $x$-coordinate
of position O4 is unrealistic.

  * b

Such heterovalent substitution looks improbably; we considered this
opportunity for the test of literature data. It worsens refinement parameters;
the results are unrealistic.

  * c

In this model, the full matrix refinement is possible; moreover, with the
simultaneous refinement of populations and ADP for incompletely occupied or
shifted positions.

These model results were compared against each other using the criterion of
refinement stability and factors R1 and wR. It turned out that for the
localization of dopant atoms, the extrema of difference Fourier synthesis,
$\Delta\rho_{max}/\Delta\rho_{min}$ are more informative. This criterion is
usually considered as the second main one for the refinement of structural
models. The main superconducting phase was determined in Model 3 with
significant reduction of $R$-factors for the same number of Bragg reflections
(152) and refinement parameters (18) as with Models 1 and 2. The highest
residual peaks of difference electronic density are localized near heavy
cations. The details of the refinement are given in Table 6.

Table 6: Details of structural refinement for YBa1.4Sr0.6Cu3O6Se0.51
crystaline phase

Temperature (K) | 294
---|---
Crystal system | Orthorhombic
space group, $Z$ |  $Pmmm$, 1
$a,b,c$ (Å) | 3.84359(4), 3.83295(6), 11.47722(17)
$V$ (Å3) | 169.092(4)
$Dx$ (g/cm3) | 6.8015
µ(mm-1) | 83.432
Profile fit |
Profile function | Pseudo-Voigt
No. of measured points | 20001
$R_{p}$, % | 1.32
$wR_{p}$, % | 1.92
GOF (S) | 3.14
Structure model fit |
Refinement based on F with a weighting scheme | 1/$\sigma^{2}$(F)
No of Bragg reflections, parameters | 152, 18
$R1$, %, | 0.87
$wR$, % | 0.73
$\Delta\rho_{\max}/\Delta\rho_{\min}$, e/Å3 | +0.17 / -0.30
Software | HighScore Plus, Jana2006

###  5.4 Processing of heat capacity data

Assuming that (at low temperatures) the heat capacity in our samples has three
major contributions, we can write

$C(T)=C_{el}(T)+C_{ph}(T)+C_{Sch}(T),$ (5)

where $C_{el}(T)$ stands for the electronic contribution containing both the
normal part $\gamma T$ and the superconductor part, $C_{ph}(T)=\beta T^{3}$
characterizes the phonon contribution, and the last term is the Schottky
anomaly, which, in case of the simplest two-level gap $\Delta_{tl}$ can be
represented as [76]

$C_{Sch}(T)=R\left(\frac{\Delta_{tl}}{T}\right)^{2}\frac{\exp\left(\Delta_{tl}/T\right)}{\left[1+\exp\left(\Delta_{tl}/T\right)\right]^{2}}.$
(6)

At the first stage, the phonon contribution was subtracted from the
experimental data333It is worth to mention that our heterophase substance
contains “parasitic” contributions to the heat capacity of the superconducting
Phase 1. At $T=15$ K, the $C/T$ of major phases $2-4$ are (in mJ/(K2 g-at)
units): $\sim 1$ for BaSeO4 [77], $\sim 0.52$ for Y2Cu2O5 [78], and $\sim 20$
for Y2BaCuO5 [79]. These exceed the contribution of the superconducting phase
(Fig. 5a). However, their contribution is mainly phononic in origin, and
subtraction of $\beta T^{2}$ from $C/T$ extinguishes it as well as the
contribution of the minor phase BaCu(SeO3)2.. For that task, $C(T)/T$ was
plotted as a function of $T^{2}$. Since the Schottky contribution (6) is
negligible (exponentially small) at $T=0$, the inclination of $C(T)/T$ near
the origin of coordinates provides the value of $\beta$ (in our case,
$\beta\approx 0.04$ (Fig. 6). Also, the value of $C(T)/T$ at $T=0$ equals
$\gamma$. After the subtraction of phonon contribution, the deviation from
typical electronic contribution of superconducting state becomes evident (cf.
Fig. 6c-e).

The characteristic superconducting behavior is restored by the subtraction of
the Schottky contribution (6). After these operations are performed, one can
make the second iteration, which involves computer coding. In the first step
in this code, the Schottky curve is subtracted from the experimental data in
accordance to (5) while allowing the opportunity to vary the gap $\Delta_{tl}$
and the prefactor $R$ around the values obtained at the initial stage. Then,
the code computes the corresponding values of $\beta$ and $\gamma$. These
values are not too far from the ones determined at the initial stage, but
performing the iteration is useful for ensuring better data processing for
further analysis. The curves in Fig. 6d were obtained via this procedure from
Fig. 6c.

For the theoretical modeling (Fig. 6b), we used the standard expression for
heat capacity [42], which, in dimensionless form, can be represented as:

$\begin{split}\frac{C_{el}(t)}{T_{c}}=\frac{1}{4t^{2}}&\int_{0}^{2\pi}d\varphi\int_{0}^{\pi}d\theta\sin\theta\\\
&\int_{-\infty}^{\infty}dx\frac{x^{2}+\delta(t)^{2}-t\delta(t)\left[d\delta(t)/dt\right]}{\cosh^{2}\left[\sqrt{x^{2}+\delta(t)^{2}}/\left(2t\right)\right]},\end{split}$
(7)

where: $T/T_{c}\equiv t;$ $\xi/T_{c}\equiv x;\Delta(T)/T_{c}\equiv\delta(t);$
$\varepsilon/T_{c}\equiv\sqrt{\xi^{2}+\Delta^{2}}/T_{c}=\sqrt{x^{2}+\delta^{2}}.$
In (7), one should substitute the value of the order parameter (the gap),
which has the form $\Delta=\Delta_{0}$ for isotropic $s-$wave and
$\Delta=\Delta_{0}\cos n\phi$ for line nodes ($n=2$ for $d-$wave
$\alpha-$model) [41, 80]. The temperature dependence of $\Delta(T)$ was
defined via the analytical expression [81] for BCS solution:

$\displaystyle\frac{\Delta}{\Delta(0)}$ $\displaystyle=$
$\displaystyle\sqrt{1-t}(0.9663+0.7733t),\text{if }t>0.33$ (8)
$\displaystyle=$ $\displaystyle 1\text{ otherwise. }$

Computation of $C_{el}(T)/T$ based on (7) and (8) was performed numerically
both for the $s-$wave and the $d-$wave cases. The best result is shown in the
Fig. 3b. It corresponds to the Schottky parameters $\Delta_{tl}=10.5\pm 0.1$
K, $R=0.31\pm 0.01$. This value of $\Delta_{tl}$ is close the Schottky gaps in
YBCO materials [82]. This matching justifies application of the simple
relation (6) instead of more sophisticated expression [83]:

$C_{Sch}=R\frac{dE_{Sch}}{dT},$ (9)

where

$E_{Sch}=\sum E_{i}\exp(-E_{i}/T)/\sum\exp(-E_{i}/T),$ (10)

and $E_{i}=g\mu_{B}M_{J}H_{eff}$ where $M_{J}=\\{-S,-S+1,...S-1,S\\}_{i}$ ,
$H_{eff}=\sqrt{H_{0}^{2}+H_{ext}^{2}}$ where $H_{0}$ and $H_{ext}$ are the
internal and external field values. In the case of double levels ($S=1/2$),
(9) transforms into (6) and the value of $\Delta_{tl}$ is expressed by the
external and the internal fields $H_{ext}$ and $H_{0}$. In the case of $S=5/2$
(which corresponds to the $2+$ states of copper ions), we obtained that the
same level of matching as in Fig. 6b is possible at somewhat lower values of
$H_{0}$. Whichever case is more relevant to our material requires further
analysis. Also, the physical cause of the Schottky anomaly in our case remains
uncertain: in addition to scattering on magnetic clusters, the scattering on
minor amount of rare earth impurities [82] is an alternative. However, the
details of Schottky anomaly are not significant for our main results, and we
will not dwell more on this topic.

##  6 Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  7 Code availability

Full code that support the findings of this study are available from the
corresponding author upon reasonable request.

##  8 Acknowledgments

A.D. acknowledges support by the Ministry of Science and Higher Education of
the Russian Federation within the State assignment FSRC ”Crystallography and
Photonics” RAS in part of X-rays diffraction study. The work of the Chapman U.
research team is supported by the US Office of Naval Research Grants No.
N00014-21-1-2879 and N00014-20-1-2442. We are grateful to E. Vinogradova for
assistance.

##  9 Author contributions

A.G., V.N. , V.G. and S.N. designed research; A.G., V.G., V.N., J.K., A.M.,
J.C., I.P, R.D., S.T., T.H. and S.C. performed research, A.G., V.N., V.G.,
A.D., R.D., S.T. and S.C. analyzed data; and A.G., V.G., A.D., R.D., S.T. and
S.C. wrote the paper.

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