# Ferromagnetic exchange field stabilized antiferromagnetic ordering in a
cuprate superconductor

Biswajit Dutta and A. Banerjee  UGC-DAE Consortium for Scientific Research,
University Campus, Khandwa Road, Indore-452001, India.

###### Abstract

We report experimental evidence of formation of antiferromagnetic clusters
well within cuprate superconductor $La_{1.85}Sr_{0.15}CuO_{4}$ (LCu) in a
composite made of LCu and ferromagnet $La_{0.6}Sr_{0.4}CoO_{3}$ (LCo). It is
found that the exchange field of LCo suppress the dynamic antiferromagnetic
spin fluctuation of LCu and short range ordered superparamagnetic type
antiferromagnetic (AFM) clusters are formed at the cost of superconducting
volume fraction. With the help of linear and non-linear ac-susceptibility
measurement we show the evidence of thermal blocking of these
antiferromagnetic clusters. Further, we show that the shrank superconducting
volume fractions undergo quantum size effect (QSE) and follow DeGennes-Tinkham
theory on finite size effect of superconductor. These give clear indication
that antiferromagnetic spin fluctuation can be a mediator of electron pairing
in cuprate superconductor.

###### pacs:

75.47.Lx, 71.27.+a, 75.40.Cx, 75.60.-d

##  I INTRODUCTION

One of the long lived mystery associated with high temperature cuprate
superconductor is the origin of coherent electron pairing in these compounds.
The magnetic state of a parent cuprate is 3D Neel antiferromagnet (AFM). After
doping holes or electrons the ordered antiferromagnetic state destroys and
superconducting state emerges [1]. However, existence of AFM type spin
correlation is reveled throughout the hole or electron doped phase diagram
(including the doping region where superconductivity exists) and depending on
dynamical nature of the AFM correlation the strength of superconductivity is
decided [1, 2, 3, 4, 5, 6, 7]. It is also observed that when the
superconducting state is destroyed by doping or by applying very high magnetic
field, then emergence of AFM like state is observed. Like Neodymium (Nd)
doping in $La_{1.85}Sr_{0.15}CuO_{4}$ (LCu) destroys the superconducting state
and results in antiferromagnetic type CDW phase [8, 9, 10]. Further,
suppression of superconductivity and emergence of Charge Density Wave (CDW)
phase or AFM state is also reveled in ferromagnet (FM) and cuprate
superconductor heterostructures at normal condition (or without imposing any
extreme condition or doping in the superconductor) [11, 12, 13, 14, 15, 16,
17, 18]. These characteristic behavior indicates a close correlation between
AFM ordering and coherent electron pairing mechanism in cuprate
superconductors, which has also opened up new perspective to understand the
superconducting pairing mechanism of cuprate superconductors. Therefore, a
detailed study of cuprate superconductor in proximity to FM systems are
required to get further knowledge about the involvement of antiferromagnetic
type spin correlation in the pairing mechanism. Mostly, such studies on oxide
superconductor and ferromagnet (SC/FM) are based on YBa2Cu3O7 (YBCO) and
La0.7Sr${0.3}$MnO3 (or La0.7Ca${0.3}$MnO3). However, the problem associated
with these YBCO based SC/FM heterostructures is high probability of disruption
of CuO chain (which act as charge reservoir in YBCO) across the interface. As
a result of that the hole concentration of YBCO gets modified, changing the
physical property in bulk like manner unrelated to interface physics [19, 20,
21, 22]. It has been observed that the SC/FM interfaces consisting of
$La_{1.85}Sr_{0.15}CuO_{4}$ (LCu) as superconductor does not show any dominant
charge transfer or orbital reconstruction phenomena to take place across the
interface [23, 24, 25, 26] and maintained their charge state intact with
respect to the parent ingredient. Hence, LCu based (or LCu type i.e.
Pr1-xCexCuO4 [23]) FM/SC composites (or bilayer) are ideal to study the
effects of exchange field [23], as well as the interfacial strain effect on
the magnetic properties of cuprate superconductors [23, 24, 25, 26].

Here, we present a study on composites made of superconductor LCu and a
ferromagnet La0.6Sr0.4CoO3 (LCo) to explore the effect of exchange bias field
on the hole doped cuprate superconductor. LCo is chosen as a ferromagnetic
counterpart because it shows long-range ferromagnetic ordering and large
magneto-crystalline anisotropy [27, 28, 29, 30], which can facilitate
substantial amount of exchange bias field on the spins of copper atom of LCu.
In this respect, it has an advantage over manganites (La0.7Sr0.3MnO3,
La0.7Ca0.3MnO3 [24, 25, 26]) to explore the effect of exchange field on
cuprate superconductors. We show that the dynamic antiferromagnetic spin
fluctuation of LCu [8, 9, 10, 31, 32, 33] is suppressed due to the magnetic
exchange field of LCo and short range order AFM phase is developed at the cost
of superconducting volume fraction. As a result, the bulk superconducting
region shrank to finite size clusters, and quantum size effect (QSE) of the
superconductor appears, without any reduction in the crystallite size of the
superconductor. The exchange field amplitude on LCu is tuned by changing the
effective interface as well as by dc magnetic field. The effective interface
is varied by following several processes like, reducing (or increasing) the
particle size, decreasing (or enhancing) the concentration of LCo in the LCu
matrix [34] and grinding the composite pallet.

We have used Linear and nonlinear ac-susceptibility measurements to study the
proximity effect induced change of the magnetic state of a superconductor in a
SC/FM interface. Nonlinear susceptibilities are found to be very effective
tool in determining various characteristics of a type-2 superconductor, such
as to identify the onset point of irreversible flux motion, nature of flux
dynamic and also very effective in determining various critical
thermodynamical parameters (like critical current, critical field and critical
temperature) [35, 36, 37, 38]. This technique is also very effective in
unambiguously distinguishing various metastable states like spin-glass,
cluster-glass and superparamagnet [39, 40, 41, 42] and also in establishing
the nature of magnetic ground state (like ferromagnetic or antiferromagnetic)
[43, 44].

In composite (or in heterostructure) the interplay between different
electronic ground states are modulated through the extended interface effect,
but in case of oxide heterostructures oxygen stoichiometry plays a big role in
modifying the physical property which is unrelated to interface physics. Hence
it is very important to determine the oxygen stoichiometry (which defines the
spin state) of the ingredients with respect to their parent compounds for a
conclusive discrimination between the phenomena related to interface effect
and the phenomena associated with the degradation of oxygen stoichiometry. We
have proposed a method to determine the spin state of a ferromagnet and hole
concentration of a superconductor in a SC/FM composite system by using the low
field magnetic ac-susceptibility technique and XRD measurement, which is
already published in Ref.[34] and briefly discussed here.

##  II SAMPLE PREPARATION AND CHARACTERIZATION

The parent ingredient LCu and LCo is prepared by pyrophoric method. Then the
composites are prepared by solid state reaction method, preparation technique
is more elaborately explained in Ref.[34]. The precursor of LCo just obtained
after pyrophoric chemical reaction is annealed at different temperatures, viz.
$950^{0}C$, $900^{0}C$ and $850^{0}C$ to obtained three different crystalline
size (CS) which are nomenclatured as LCo950, LCo900 and LCo850 respectively
((CS)LCo850$<$(CS)LCo900$<$ (CS)LCo950). Then 76 weight percentage of LCu and
24 weight percentage of LCo950 is mixed, palletized at comparatively high
pressure (i.e., 150 kilo-Newton (kN)) and heated at 800${}^{0}C$ to prepare
composite A1. In the similar way A2 and A3 is prepared by mixing LCo900 and
LCo850 with LCu respectively. The concentration of LCo850 is also varied in
LCu matrix to change the effective interface between LCu and LCo. In this case
LCu and LCo850 is mixed at different weight ratio like 76:24, 85:15 and 95:5
and these composites are nomenclatured as A3, A4 and A5, respectively. The
parent ingredients are independently grinned palletized at similar pressure
(i.e., 150kN) and annealed at same temperature (i.e. 800 0C) for comparison
purposes. The weight percentage detail of the corresponding parent ingredient
present in the composites (i.e., A1, A2, A3, A4, A5) and detailed nomenclature
of all composites along with the crystalline size of parent LCu and LCo are
provided in TABLE I. The crystalline size of the ferromagnet and the
superconductor is calculated from the XRD pattern as well as from Transmission
electron microscopy (TEM) measurement.

Table 1: List of composites and crystalline size details Composite Name |  LCu [Weight%] + LCo (annealing Temperature) [Weight%] |  average crystalline Size of LCo (nm)
---|---|---
A1 |  LCu[76%]+LCo(950)[24%] |  69(nm)
A2 |  LCu[76%]+LCo(900)[24%] |  47(nm)
A3 |  LCu[76%]+LCo(850)[24%] |  27(nm)
A4 |  LCu[85%]+LCo(850)[15%] |  27(nm)
A5 |  LCu[95%]+LCo(850)[5%] |  27(nm)

###  II.1 Structural Characterization : XRD

X-ray diffraction (XRD) measurements are performed in Bruker X-ray
diffractometer from $10^{0}$ \- $90^{0}$ at an interval of $0.02^{0}$. FIG.
1(a) shows the two-phase Rietveld refinement of the composite A3 and FIG. 1(b)
shows the XRD pattern of A1, A2 and A3 respectively. FIG. 2(a) shows the
Rietveld refinement of the composite A5 and the combined plot of the XRD
spectrum of A3, A4 and A5 composite are shown in FIG. 2(b) for comparison
purpose. The phase fractions and other structural details obtained from two
phase Rietveld refinement are given in TABLE II.

Figure 1: (Colour Online) (a) Rietveld refinement of the composite A3. (b) XRD plot of all composites A1, A2, A3 (Inset shows the region $30^{0}$-$35^{0}$ shows the maximum intensity peak region) Table 2: Best fitted parameters, SG-Space group, PF-Phase fraction, a, b and c are lattice parameters Material | SG | a(A${{}^{0}})$ | b(A${{}^{0}})$ | c(A${{}^{0}})$ | PF | $R_{f}$
---|---|---|---|---|---|---
(LCu) | I4/mmm | 3.78(9) | 3.78(9) | 13.20(5) | NA | 1.35
(LCo850) | R-3c | 5.44(5) | 5.44(5) | 13.22(5) | NA | 1.44
(LCo900) | R-3c | 5.44(6) | 5.44(6) | 13.21(7) | NA | 1.44
(LCo950) | R-3c | 5.44(3) | 5.44(3) | 13.20(5) | NA | 1.44
A1(LCu) | I4/mmm | 3.78(9) | 3.78(9) | 13.20(3) | 76 | 1.47
A1(LCo) | R-3c | 5.44(2) | 5.44(2) | 13.20(9) | 24 | 1.47
A2/( LCu) | I4/mmm | 3.78(3) | 3.78(3) | 13.20(9) | 76 | 1.44
A2/( LCo) | R-3c | 5.44(6) | 5.44(6) | 13.21(5) | 24 | 1.44
A3/( LCu) | I4/mmm | 3.78(2) | 3.78(2) | 13.20(3) | 76 | 1.8
A3/( LCo) | R-3c | 5.44(1) | 5.44(1) | 13.21(7) | 24 | 1.8
A4/( LCu) | I4/mmm | 3.78(4) | 3.78(4) | 13.20(8) | 86 | 1.6
A4/( LCo) | R-3c | 5.44(9) | 5.44(9) | 13.21(9) | 14 | 1.6
A5/( LCu) | I4/mmm | 3.78(6) | 3.78(6) | 13.20(9) | 95.9 | 1.87
A5/( LCo) | R-3c | 5.44(8) | 5.44(8) | 13.22(1) | 4.1 | 1.87
Figure 2: (Colour Online) (a) Rietveld refinement of the composite A5. (b) XRD
plot of all composites A3, A4, A5.

The lattice parameters of LCu obtained after the two phase Rietveld refinement
of all composites (i.e., A1 to A5) do not show any significant change, which
indicates the c/a ratio or the Orthorhombic distortion of LCu remain same for
all composites. This indicates that there is no change in hole concentration
of LCu [45]. The crystalline size of LCu and LCo is calculated using
Williamson-Hall (W-H) analysis. The hole concentration of parent LCo is
determined by Iodometric titration method. The chemical formula obtained from
these results are as follows; like for LCo950, LCo900 and LCo850 it is
$La_{0.6}Sr_{0.4}CoO_{2.97}$, $La_{0.6}Sr_{0.4}CoO_{2.99}$ and
$La_{0.6}Sr_{0.4}CoO_{3.01}$ respectively. From this chemical formula the
concentration ratio of Co+3 and Co+4 ions can be calculated very easily and
this ratio is further used to calculate the theoretical values of effective
magnetic moment ($\mu_{eff}$) and saturation moment of the corresponding LCo.
Experimentally the value of $\mu_{eff}$ has been calculated from the Curie-
Weiss fitting in the paramagnetic region of LCo and the obtained value is
compared with the theoretical effective magnetic moment value of
LCo($(\mu_{eff})_{expt}$=4.33$\pm$0.01). As LCu is a Pauli paramagnet and the
moment value above superconducting onset temperature is found around 10-6 emu
(much smaller than the Curie-Weiss moment value of LCo), and also found
constant of temperature up-to room temperature. Therefore, the effective
magnetic moment value of parent LCo and the corresponding composites has to be
same provided there is no change of hole concentration or chemical reaction.
We have used this concept to determine the spin state (i.e. hole
concentration) of LCo present in the composite [34].

###  II.2 Transmission electron microscopy (TEM)

Transmission electron microscopy (TEM) image of composite A3 is shown in FIG.
3(a). The high resolution transmission electron microscopy (HRTEM) image is
shown in FIG. 3(b), FIG. 3(c) and FIG. 3(d). The high resolution images are
depicting that the crystallites of LCo are connected and combined with the
crystallite of LCu, respectively. The fringes of the marked lattice spacing of
0.5 nm, corresponds to LCo. FIG. 3(d) indicates very sharp interface, which
clearly depicts there is no intermixing between LCu and LCo. We have performed
similar TEM and HRTEM measurements at various places of A3 and also performed
High angle annular dark field imaging (i.e., elemental scan), ( see Section. A
in the Supplemental Material below for more elaborate explanation).

Figure 3: (Color Online) HRTEM image of composite A3 is shown at various
resolution.

##  III EXPERIMENTAL DETAILS

The Low field linear and nonlinear magnetic ac-susceptibility measurements
have been performed using a homemade ac-susceptometer, which can be operated
down to 4 K from 300 K and the measurements can be done in both cooling and
heating cycle with a temperature accuracy better than 1 mK. The estimated
sensitivity of the setup is $\sim 10^{-7}$ emu [46]. The higher dc-field($>$
200 Oe) superimposed ac-susceptibility measurements are performed in MPMS-XL
(M/S, Quantum Design).

###  III.1 Ac-susceptibility measurement

The magnetic property of the composite has been studied by extensive use of
ac-susceptibility measurement which is capable to probes the spin dynamic at
very low field. The magnetization (m) can be expanded with respect to the
applied ac-field $h_{ac}(h)$ as

$m=m_{0}+\chi_{1}h+\chi_{2}h^{2}+\chi_{3}h^{3}+\chi_{4}h^{4}....(1)$

$\chi_{1}(\approx\delta m/\delta h$) is linear susceptibility and $\chi_{2}$,
$\chi_{3}$, $\chi_{4}$… are nonlinear susceptibilities. These nonlinear
susceptibilities contain many fruitful information but magnitude of these are
much smaller (couple of order) than the linear susceptibility, therefore they
are difficult to measure from normal dc magnetization measurement, but these
nonlinear susceptibilities can be easily measured from high sensitive ac-
susceptibility measurement [47]. If the magnetization has an inversion
symmetry with respect to the applied ac-field ($h_{ac}$) then all the even
order susceptibilities, like $\chi_{2}(\approx\delta^{2}m/\delta^{2}h$),
$\chi_{4}(\approx\delta^{4}m/\delta^{4}h$) are zero without any externally
applied dc-field ( i.e. at hdc = 0 Oe) like paramagnetic and antiferromagnetic
materials, but when the inversion symmetry breaks w.r.t the sign of $h_{ac}$
then $\chi_{2}$, $\chi_{4}$.., all the even order susceptibility shows finite
value (at hdc = 0 Oe), like in case of ferromagnet, or ferrimagnetic state[47,
48, 49, 50].

Third order susceptibility ($\chi_{3}$) is very useful tool to discriminate
between various metastable states (like spin glass (SG), superparamagnet (SPM)
etc.) ably [51, 40, 41, 52, 53]. For example, if there are inter-particle
interaction then the frequency dependent nature of $\chi_{1}^{R}$(T) for a SPM
system will give similar result as that of a SG system. Hence, in this type of
situation it becomes difficult to discriminate between SG and SPM by looking
only at the frequency dependent $\chi_{1}^{R}$(T) result [52]. Whereas across
spin glass transition a negative cusp in $\chi_{3}$ is observed which is
correlated with the divergence of Edward-Anderson order parameter, but in case
of SPM this type of divergence is absent [51, 40, 41]. $\chi_{3}$ is also
found as very effective tool in determining the universality class of the
ferromagnet i.e. to study the nature of magnetic ground state of the
corresponding ferromagnet [48, 54]. Therefore, the behavior of $\chi_{3}$
around ferromagnetic to paramagnetic transition (TC) is very important in
deciding the nature of magnetic interaction or the nature of magnetic ground
state of a ferromagnet.

##  IV Results and discussions

The combined temperature-dependent plot of mole normalized $\chi_{1}^{R}$(T)
(the real part of first order susceptibility) of A1, A2 and A3 as well as
their respective ferromagnetic constituent is shown in FIG. 4(a).
$\chi_{1}^{R}$(T) of the composite A1 is indicated by down red triangle and
$\chi_{1}^{R}$(T) plot of the corresponding ferromagnet LCo950 is indicated by
Red line. Similarly black and blue color (symbols and line) describes the same
for composite A2 and A3 respectively. All measurements are performed in
heating cycle. Normalization of $\chi_{1}^{R}$(T) is performed with respect to
the mole fraction of LCo as obtained from the Rietveld refinement of the XRD
patterns [34]. The Zoomed view of $\chi_{1}^{R}$(T) around FM TC is shown in
the upper inset of FIG. 4(a), it is observed that around TC and above it the
$\chi_{1}^{R}$(T) value of the composite matches with the $\chi_{1}^{R}$(T)
value of the corresponding ferromagnet present in that. These observation
indicates that the Curie temperature ($\theta_{C}$) and effective magnetic
moment ($\mu_{eff}$) of the composites and the corresponding parent LCo are
same ($\theta_{C}$ and $\mu_{eff}$ have been calculated from the Curie-Weiss
fitting of the paramagnetic region of $\chi_{1}^{R}(T)$ graph [34]). It
suggests the magnetic structure (or oxygen stoichiometry) of LCo in the
composites remain same with respect to their parent LCo. Similar conclusion
has been drawn from $\chi_{3}$ as shown later (FIG. 5). Hence the anomaly
around the superconducting onset temperature, $T_{S(onset)}$, shown in the
lower inset of FIG. 4(a) indicates the modification of magnetic state of LCu
due to close proximity of LCo. The anomaly is more prominent in case of A3
than A2 and A1, because A3 consists of smaller ferromagnetic particle LCo850,
hence it has larger surface to volume ratio than LCo900 and LCo950, due to
this reason the effective interface between LCu and LCo is larger in case of
A3 than A1 and A2. Therefore the above observation depicts the anomaly around
TS(onset) is an interface effect.

Figure 4: (Color Online) (a)The combined temperature dependent plot of
$\chi_{1}^{R}(T)$ for the composite A1 (red triangle) and LCo950 (Red line),
composite A2 (black circle) and LCo900 (black line), composite A3 ( blue
rectangular box) and LCo850(blue line). Upper inset shows the zoomed view
around ferromagnetic to paramagnetic transition temperature(TC). Lower inset
shows the zoomed view around superconducting onset temperature (TS(onset)).
(b) The combined temperature dependent plot of A3, A4, A5, LCu and LCo
depicted. All the measurements are done in an ac-field of amplitude 3 Oe and
frequency 231.1 Hz.

To cross check the statement related to the interface effect, the ratio of LCu
and LCo is changed in composites A4 and A5 (as mentioned before) to vary the
number of LCo grains sprinkled across LCu domain, which also changes the
effective interface between LCu and LCo. FIG. 4(b) shows the comparative
normalized $\chi_{1}^{R}$(T) plot of composites A3, A4 and A5, along with them
the normalized $\chi_{1}^{R}$(T) graph of parent LCo and LCu is also shown for
comparison purpose. The onset temperature of superconductivity
($T_{S(onset)}$) of LCu is observed around 32 K and above $T_{S(onset)}$,
$\chi_{1}^{R}(T)$ shows Pauli paramagnetic behavior. The paramagnetic to
ferromagnetic transition temperature (TC) of LCo850 is observed around 233 K.
The anomalous hump in $\chi_{1}^{R}$(T) around the superconducting transition
gets suppressed with the reduction of LCo850 concentration in LCu matrix
(shown in FIG. 4(b)), like for A4 the amplitude of the peak value suppresses
and become almost flat for A5. From Curie-Weiss fitting in the paramagnetic
region the value of $\mu_{eff}$ and $\theta$ is obtained around 4.28 $\mu_{b}$
and 233 K respectively. The Hopkinson peak amplitude (i.e the peak in
$\chi_{1}^{R}$(T) graph just below FM TC) for all the composites remain same,
which indicates the magnetic domain size of LCo is same in all composites.
Thus, the observed anomaly suppresses in similar manner as that of FIG. 4(a),
which is an indication that the anomaly in $\chi_{1}^{R}$(T) measurement is an
interface driven phenomena.

Figure 5: (Color Online) The temperature dependent mole normalized Real (RL)
and Imaginary (IM) parts of $\chi_{3}$ of Composites A3 and LCo850. Inset
shows the zoomed view of the anomalous region around superconducting onset
(TS(onset)). The measurements are performed at ac-field($h_{ac}$)= 3 Oe and
frequency(f)= 231.1 Hz.

The mole normalized $\chi_{3}$ plot of the composite A3 and the corresponding
ferromagnet is shown in FIG. 5 (the number of mole of LCo present in the
composite is used to normalize the $\chi_{3}$ graph of the composite). It
shows that the nature of $\chi_{3}$ (both real and imaginary parts) remain
same for the composite and the ferromagnet except around the superconducting
onset temperature (TS(onset)), which gives direct indication that the magnetic
state of the ferromagnet present in the composite remain same as that of the
parent ferromagnet.

The interface effect is further verified by measuring the $\chi_{1}^{R}(T)$
behavior of a composite pallet (A2) and the powder specimen of same composite;
the comparative plot is shown in FIG. 6. The inter-granular connection reduces
after making powder of the composite pallet and hence the effective interface
between LCu and LCo reduces. The difference between the $\chi_{1}^{R}(T)$ plot
of powder and pallet around $T_{S(onset)}$ can be clearly distinguished from
the inset of FIG. 6. For powder the hump is suppressed as a result of
reduction of effective interface but there is no change in the critical
temperature of the ferromagnet. So all these measurements are unambiguously
show that when the effective interface reduces then the anomaly around
$T_{S(onset)}$ also decreases, which clearly proves the phenomena is an
interface effect.

Figure 6: (Color Online) The plot of $\chi_{1}^{R}(T)$ against temperature for
A2 composite in two different conditions. The red square block shows the
result for pallet and the black circle for the powder samples. Inset shows the
zoomed view of the anomalous region. The measurements are performed at ac-
field($h_{ac}$)= 3 Oe and frequency(f)= 231.1 Hz.

The gradual decrease of the diamagnetic fraction below $T_{S(onset)}$ while
increasing the effective interface is evident from FIG. 4(b). The crystallite
size and the lattice parameter of LCu (and LCo) remain almost same for all
composites (evident from the XRD graph). The magnetic ac-susceptibility
measurement depicts the demagnetization factor and spin state of LCo also
remain same for all the composites (evident from the equal values of the
Hopkinson peak height at 226 K and paramagnetic moment), so the decrement of
diamagnetic phase fraction is not related to the degradation of hole
concentration or structural distortion of either LCu or LCo. This indicates a
possibility that there has developed some kind of magnetic interaction between
LCu and LCo across the interface, which causes the observed changes in the
magnetic state of LCu (like, the decrease in diamagnetic susceptibility and
excess susceptibility across $T_{S(onset)}$). Otherwise, if it is due to the
change of magnetic state of LCo then the change could have been observed
around TC with increasing or decreasing concentration of LCo (in FIG. 4(b)).
Along with that, above TS(onset) the magnetization strength of LCu (i.e. Pauli
paramagnetic) is much smaller than LCo (ferromagnetic), which can not affect
the magnetic state of LCo but the vice versa is highly probable. Hence, it can
be concluded that the magnetic state of LCu is getting affected because of its
proximity to LCo, without any change in the chemical composition of the
constituents in the composites.

To reveal the nature of perturbed magnetic state of LCu and also the type of
magnetic interaction across the interface, the higher order (i.e. second
order-$\chi_{2}$ and third order-$\chi_{3}$) ac-susceptibility measurements
are performed because of their uniqueness to distinguish between various
magnetic state (as previously discussed). Around TS(onset), $\chi_{3}^{R}$
shows a dip like feature (shown in the inset of FIG. 5) and around the similar
temperature range $\chi_{1}^{R}$ shows hump like behavior (as shown in FIG.
4). This behavior is evocative of $\chi_{3}^{R}$ as observed in spin glass
(SG) [40, 51, 55] and superparamagnetic (SPM) systems [41, 56, 57]. But unlike
SG in this case $\chi_{3}^{R}$ doesn’t show any critical behavior with respect
to applied magnetic field, frequency and temperature. Moreover, the
temperature dependent behavior of $\chi_{1}$ and $\chi_{3}$ follow Wohlfarth
model [56, 57, 52, 41]. The ac-field dependent behavior of
$\arrowvert\chi_{3}^{R}\arrowvert$ is shown in the right hand side upper inset
of FIG. 7(a) and the value of $\arrowvert\chi_{3}^{R}\arrowvert$ approaches
towards saturation with decreasing the amplitude of ac-field, which is a clear
indication of the blocking phenomena of magnetic clusters. According to
Wohlfarth SPM model [56] the magnetization(M) of a non interacting single
domain SPM particle is represented as

$M=n\bar{\mu}L(\frac{\bar{\mu}}{k_{B}T})$ (1)

where n is the number of particles per unit volume, $\bar{\mu}$ is the average
magnetic moment of a single magnetic entity or particle, $k_{B}$ is the
Boltzmann’s constant, and L(x) is the Langevin function. So, from Eqn. 1 the
linear susceptibility can be expressed as

$\chi_{1}^{R}=\frac{\bar{\mu}}{3k_{B}T}$ (2)

and the third order susceptibility is represented as

$\chi_{3}^{R}=-(\frac{n\bar{\mu}}{45})(\frac{\bar{\mu}}{k_{B}T})^{3}$ (3)

Figure 7: (Colour Online) (a) $\arrowvert\chi_{3}^{R}\arrowvert$ is plotted
against $T^{-3}$ and the black line shows the linear fitting. Inset (Top
Right) $\arrowvert\chi_{3}^{R}\arrowvert$ is plotted against amplitude of
applied ac-magnetic field at temperature 33 K for the sample A3. Inset (Down
Left) $\chi_{1}^{R}$ is plotted against $T^{-1}$ above 32K black line shows
the linear fitting to the data. (b) Imaginary part of $\chi_{1}^{I}$(Right
hand side Y axis) and $\arrowvert\chi_{2}\arrowvert$ (left hand side Y axis)
are plotted against temperature of A3 composite. The measurements are
performed at $h_{ac}$ = 12 Oe at an exciting frequency of (f) = 231.1 Hz. Only
in the Top-Right inset of (a) the the exciting ac-field is varied at same
frequency.

Therefore, Eqn. 2 and Eqn. 3 demonstrates that in the SPM region
$\chi_{1}^{R}$ is positive and it follows $T^{-1}$ behavior, whereas
$\chi_{3}^{R}$ is negative and it follows $T^{-3}$ behavior [56]. FIG. 7(a)
shows the $T^{-3}$ temperature dependence for experimentally obtained
$\chi_{3}^{R}$ and lower inset of FIG. 7(a) shows the $T^{-1}$ dependency of
$\chi_{1}^{R}$ above 32 K. The value of $\bar{\mu}$ ($\sim 10^{3}$ $\mu_{b}$,
$\mu_{b}$ is Bohr magneton) is calculated from the slope ratio obtained from
the straight line fitting of $\chi_{3}^{R}$ and $\chi_{1}^{R}$. In case of a
conventional SPM cluster (smaller particles of ferromagnet) the values of
$\bar{\mu}$ are quite large (as it contains large number of ferromagnetically
align spins inside its volume, for a pure ferromagnetic clusters it is of the
order of 104-106$\mu_{b}$ [52]) compared to the number we have obtained from
fitting. It indicates that the SPM clusters are not ferromagnetic in nature.
Moreover, there is no anomaly in the temperature dependent
$\arrowvert\chi_{2}\arrowvert$ measurement is observed through out the
temperature interval as shown in FIG. 7(b) (scale: Left hand Y axis scale),
indicating zero value of internal field (i.e. non-ferromagnetic type
clusters). FIG. 7(b) also shows the temperature dependent plot of
$\chi_{1}^{I}$ (scale: right hand Y axis scale), below 33 K a hump like
features is observed in $\chi_{1}^{I}$ graph and no further anomaly in
$\chi_{1}^{I}$ is observed above 33 K. $\chi_{1}^{I}$ corresponds the
imaginary part of first order ac-susceptibility and signifies area of minor
magnetic hysteresis loop [35], which does not show any anomaly both around and
above blocking temperature TB. Therefore all these observations are ruling out
the possibility of forming ferromagnetic type of clusters around TS(onset). In
ac-susceptibility measurement at zero dc bias field the AFM type ordered
systems (or clusters) show zero value of internal field
(i.e.$\arrowvert\chi_{2}\arrowvert$$=$0) and null value of minor hysteresis
loop area (i.e. $\chi_{1}^{I}$=0) [53], which is similar to our observations.
Therefore, the fundamental and higher order susceptibility measurement
confirms the excess susceptibility or hump in $\chi_{1}^{R}$ around TS(onset)
appears due to the blocking of the SPM type AFM clusters.

Figure 8: (Color online) (a) $\chi_{1}^{R}$ data of A3 is plotted against
temperature in five superimposed dc-field 0, 100, 200, 500 and 1000 Oe (b)
$\arrowvert\chi_{2}\arrowvert$ graph of A3 is shown in three superimposed dc-
fields 20, 50 and 150 Oe, The measurements are performed in ZFC and FCW (Green
bold line) along with that the plots for LCu and LCo are also shown at 50 Oe
superimposed dc-field in ZFC mode. Inset shows the isotherm of
$\arrowvert\chi_{2}\arrowvert(H)$ of A3 at T=35 K. For all measurement the
amplitude and frequency of the ac-field are $h_{ac}$=12 Oe and f=231.1 Hz.

SPM clusters are short range ordered magnetic clusters and its anisotropy
energy is comparable to the thermal energy. Due to this reason the clusters
show dynamical behavior above blocking temperature but the microscopic spin
structure or the interaction between the spins remain same to that of long
range order magnetic structure or state [41]. These short range ordered AFM
clusters can appear from the covalent bonding between $Cu^{+2}$ ions and
$Co^{+3}$ or $Co^{+4}$ ions across the interface, as it is observed in
YBCO/LCMO heterostructure [14, 15]. But in our case the large suppression of
the diamagnetic fraction (shown in FIG. 4(b)) along with the significant
change of $\chi_{1}^{R}$(T) around $T_{S(onset)}$ ($\sim 10^{-5}$ emu, shown
in FIG. 4) indicates the perturbation is not only confine to the interface but
it is propagated well inside the bulk of LCu i.e. the bulk magnetic property
of LCu is modulated due to close proximity of LCo. Recently, the emergence of
bulk CDW phase is evident in cuprate/ferromagnet heterostructure [16, 17] and
the appearance of similar CDW phase, AFM phase, AFM glass phase is also
evident while the superconductivity of a cuprate is destroyed by applying very
high magnetic field or by applying very high strain on it [8, 9, 10, 58, 59,
60, 61]. Here we have observed formation of the short range ordered SPM type
AFM clusters along with decrement of diamagnetic fraction which are unrelated
to change in crystal structure or change of charge carrier concentration.
Hence, the most probable reason is, the AFM fluctuation of LCu is suppressed
by the exchange magnetic field of LCo. The exchange bias field can propagate
through the covalent bonding between $Cu^{+2}$ ions and $Co^{+3}$ or $Co^{+4}$
ions across the interface or directly inside LCu, and resulted in formation of
short range order SPM type AFM clusters both across the interface and also
within the bulk of LCu.

The presence of AFM type spin fluctuation in optimally doped LCu has already
been found from various measurements like specific heat, neutron diffraction,
X-ray scattering [1, 2, 3, 7, 8, 9, 10, 31, 32, 33] etc. In this section we
are going to emphasize on the phenomena related to exchange magnetic field
stabilized AFM ordering in cuprate superconductor. For that a small dc bias
field is superimposed with the ac-field during the susceptibility
measurements. FIG. 8(a) shows the dc-field superimposed $\chi_{1}^{R}$(T) plot
of composite A3, the peak value of $\chi_{1}^{R}$(T) suppress and the peak
temperature (corresponding to the blocking temperature TB) shifts towards
higher temperature ($\Delta$(TB) $\sim$ 6 K) with increasing the amplitude of
dc bias field. The applied maximum dc bias filed (hmax) is very small in
magnitude, hence the Zeeman energy does not have sufficient strength to
disturb the already formed AFM clusters in LCu, but the magnetic response of
LCo can be manipulated. At very lower value of dc bias field, the loosely
coupled surface spins are responding and when the dc-field amplitude exceeds
the value of demagnetization field of the ferromagnet ($\sim$50 Oe), then
spins at the bulk of LCo are responding, because above the demagnetization
field Zeeman energy overcome the magneto crystalline anisotropy energy and
hence the cobalt spins align along a resultant direction decided by the
competition between these two interactions. As the cobalt spins are coupled
with the copper spins by exchange force, therefore the effective exchange bias
field on LCu created by LCo also changes due to the change of orientation of
cobalt spins. Thus, the preformed AFM clusters feel this change and get biased
indirectly by this small value of applied dc-field, and hence the decrement of
relaxation peak height of $\chi_{1}^{R}(T)$ and the positive increment of TB
with increasing the amplitude of dc bias field is observed.

To further emphasize on this exchange bias model the dc-field superimposed
second order susceptibility ($\chi_{2}$) measurements are performed, as the
application of dc bias field tunes the amplitude of effective exchange bias
field seen by the copper spins of LCu. The copper spins are at different
distances from the interface hence, each spins of LCu is going to feel
different amplitude of exchange force created by the cobalt spins. So, the
internal spin arrangement of the antiferromagnetic clusters in LCu is not
going to remain same as the previous case of hdc=0 Oe. Therefore, these change
of the internal field amplitude is going to reflect on the second order
susceptibility ($\arrowvert\chi_{2}\arrowvert$). The dc-field superimposed
temperature dependent $\arrowvert\chi_{2}\arrowvert$ measurement
($\arrowvert\chi_{2}\arrowvert$(T)) is shown in FIG. 8(b). The anomaly in
$\arrowvert\chi_{2}\arrowvert$(T) appears around 50 K and persists up to 5 K.
These measurements are performed in ZFC (zero field cooled) mode, the sample
is cooled in zero field and all measurements are performed in warming cycle in
the presence of dc-field ( $h_{dc}$= 20, 50 and 150 Oe respectively). One of
the graph in FIG. 8(b) (Green bold line) corresponds to the FCW (field cool
warming) measurements taken at $h_{dc}$= 50 Oe. In this protocol dc-field is
applied above $T_{C}$ of A3, then cooled to 5 K and measurements are performed
during warming cycle in the presence of same dc-field (50 Oe). In ZFC
measurement initially the peak value of $\arrowvert\chi_{2}\arrowvert(T)$
(FIG. 8(b)) increases up to dc-field of amplitude 50 Oe and then it decreases
with further increasing the amplitude of the dc bias field. Inset shows the
isotherm of $\arrowvert\chi_{2}\arrowvert$
(i.e.$\arrowvert\chi_{2}\arrowvert(H)$) at 35 K, revealing the same details as
obtained from $\arrowvert\chi_{2}\arrowvert(T)$, viz
$\arrowvert\chi_{2}\arrowvert$ decreases after a particular field (50 Oe).
Similar measurements are performed for the parent ingredients, but no anomaly
in $\arrowvert\chi_{2}\arrowvert$ is observed around these temperature(i.e. 20
K-50 K). The $\arrowvert\chi_{2}\arrowvert$(T) graph of LCu depicts the
appearance of $\arrowvert\chi_{2}\arrowvert$ below about 25 K (In case of
cuprate superconductor magnetic field dependent critical current density is
the source of $\arrowvert\chi_{2}\arrowvert$ [36, 37, 38]) and in case of LCo
$\arrowvert\chi_{2}\arrowvert$ appears around $T_{C}$ ($\sim$233 K, because of
the appearance of internal symmetry breaking field but much below $T_{C}$,
$\arrowvert\chi_{2}\arrowvert$ shows almost zero value because of the
domination of demagnetization effect), so neither LCo nor LCu alone is
responsible for the anomaly observed in $\arrowvert\chi_{2}\arrowvert$ around
35 K for the composites.

This anomalous variation of $\arrowvert\chi_{2}\arrowvert$ with dc-field can
be very nicely explained by using the exchange coupling model. The
mathematical form to describe exchange coupling between two magnetic material
across the interface is as follows [62, 63, 64, 65]

$H_{ex}=-j\frac{S_{AFM}\times S_{FM}}{\mu\times t_{FM}\times M_{FM}}$ (4)

$H_{ex}$ is the exchange bias field, J is the exchange integral across the
FM/AFM interface per unit area, $S_{AFM}$ and $S_{FM}$ are the interface (or
surface) magnetization amplitude of the antiferromagnet and the ferromagnet,
respectively, $t_{FM}$ is the thickness of the ferromagnetic domain and
$M_{FM}$ is the magnetization of the FM layer at a magnetic filed ‘h’. At
lower value of dc bias field ($<$50 Oe), the spins across the interface are
only going to respond (i.e. $S_{FM}$, as they are loosely coupled to the bulk
of LCo), whereas the bulk magnetization are not going to respond at this lower
value of dc-field i.e. $M_{FM}$ and $t_{FM}$ remain unchanged (because the
applied field amplitude is still lower than the values of demagnetization
field). Therefore, according to Eqn.4 at lower dc-field amplitude the
effective $H_{ex}$ (Local field) on copper spins is larger in amplitude, which
causes comparatively sizable misalignment between the AFM coupled copper spin,
as they are at different distance from the interface. Due to this reason the
increment of $\arrowvert\chi_{2}\arrowvert$ is observed up-to a certain value
of dc magnetic field (i.e. 50 Oe). On further increasing the dc-field
amplitude the bulk spins start to respond, because it exceeds the amplitude of
demagnetization field of LCo (i.e., $M_{FM}$ and $t_{FM}$ increases
simultaneously), as a result $M_{FM}$ and $t_{FM}$ start to dominant over
$S_{FM}$, and hence, $H_{ex}$ reduces in accordance with Eqn. 4. As a result
the misalignment between the copper spins reduces and decrement of
$\arrowvert\chi_{2}\arrowvert$ is observed at higher value of dc bias field.
Similarly, the difference of $\arrowvert\chi_{2}\arrowvert$(T) values between
ZFC and FCW protocol (shown in FIG. 8(b)) can also be explained by using Eqn.
4. As the ferromagnetic domain size ($t_{FM}$) and $M_{FM}$ of LCo in FCW
condition is larger than the ZFC condition (below saturation field), evident
from higher value of $\arrowvert\chi_{2}\arrowvert$ in case of FCW than ZFC
above 50 K, i.e. higher value of internal field, hence the resultant exchange
bias field values on copper spins is larger in ZFC case than FCW case. As a
result the observed values of $\arrowvert\chi_{2}\arrowvert$(T) below 50 K is
lower in FCW case than it is in ZFC case. This observation further supports
the proposed exchange bias model. Additionally, a measurement of the minor
magnetic hysteresis loop, which is discussed in the Supplemental Material (See
Section. B), also shows evidence of an exchange bias between the Copper and
Cobalt spins.

Figure 9: (Color online) (a) $\chi_{1}^{I}$ is plotted against temperature for
A2, A3 and A4. (b) $H_{C2}$ is plotted against temperature for A2,A3 and A4
composites.

These preformed SPM type AFM clusters are also observed to affect the
superconducting state of LCu. Hence, a comprehensive understanding on the
evolution of superconductivity in these composites require information on the
critical thermodynamic parameters such as critical field ($H_{C2}$) and
critical current density ($J_{C}$) of LCu. $H_{C2}$ and $J_{C}$ are affected
because of disorder and while the system size becomes comparable to certain
characteristics length scale, like coherence length ($\xi_{0}$) and London
penetration depth($\lambda_{l}$) [66, 67, 68]. Here the imaginary part of
$\chi_{1}$ ($\chi_{1}^{I}$ ) has been used as a probing tool to realize the
nature of $H_{C2}$ and $J_{C}$. In ac-susceptibility measurement the phase lag
between the ac-driving field and corresponding magnetization results in
$\chi_{1}^{I}$. In case of a type-2 superconductor $\chi_{1}^{I}$ appears
around $T_{S(onset)}$ because of vortex formation and pinning of it [69, 70,
71]. The maximum point of $\chi_{1}^{I}$ represents the temperature ($T_{S}$)
at which the bulk superconductivity vanishes i.e. the critical current density
($J_{C}$) $\longrightarrow$0 and the applied magnetic field fully penetrates
inside the superconductor [35, 69, 70, 71]. According to Bean model [35] the
hysteresis loss (W) is inversely proportional to the critical current density
($J_{C}$), hence, at T $\longrightarrow$ $T_{S}$, $J_{C}$$\longrightarrow$0,
so W diverges at $T_{S}$ , as a result of that at T= $T_{S}$ peak in
$\chi_{1}^{I}$ is observed [35-37]. The dc-field dependent behavior of
$\chi_{1}^{I}$ around TS(onset) describes the anomaly is because of
superconductivity [35] (see Section. C in the Supplemental Material). The
temperature dependent $\chi_{1}^{I}$ plot of A2, A3 and A4 are shown in FIG.
9(a). The onset point of $\chi_{1}^{I}$ for A3, A4 and A2 are observed around
34 K, 31 K and 30 K respectively, whereas the diverging temperature of
$\chi_{1}^{I}$ is observed at $\sim$24.5 K, $\sim$20 K and $\sim$19 K for A3,
A4 and A2, respectively. The crystalline size of LCu is same for all
composites (as previously discussed), therefore according to Bean model and
HC, TS and JC phase diagram of a superconductor [35, 36, 37] the maximum point
of $\chi_{1}^{I}$ is decided by the strength of the critical current density.
In case of A3 maximum point of $\chi_{1}^{I}$ is observed at higher
temperature than A2 and A4, which clearly indicates that the strength of
critical current density is highest in case of composite A3, and according to
the obtained temperature values, the amplitude of critical current density of
the three composites can be expressed in the following descending order i.e.,
$J_{C}(A3)>J_{C}(A4)>J_{C}(A2)$. Therefore, at any particular temperature
below TS(onset) the amplitude of critical field required to destroy the
superconductivity is also highest for the corresponding material having
highest value of critical current density. The plot of $H_{C2}$ against
$T_{S}$ for A3, A2 and A4 is shown in FIG. 9(b), where $H_{C2}$ can also be
written in the following descending order $H_{C2}(A3)>H_{C2}(A4)>H_{C2}(A2)$.
Therefore, all these results are indicating that the composites having larger
AFM volume fraction show smaller diamagnetic fraction (discussed previously in
FIG. 4) and largest values of $H_{C2}$ and JC. This kind of unusual increase
of the amplitude of critical thermodynamical parameters and decrease of
diamagnetic fraction (shown in FIG. 4(b)) is often observed in the Quantum
Size regime of superconductors, known as Quantum size effect (QSE). The QSE
model was proposed by DeGennes and Tinkham [66, 67, 68], where the
mathematical expression for critical field ($H_{C2}$), critical current
density (JC) and effective London penetration depth ($\lambda_{eff}$) in the
quantum limit can be represented as [66, 67]

$H_{C2}\sim\frac{\xi_{0}\lambda_{l}}{r^{1.5}}$ (5) $J_{C}\sim\frac{1}{r^{3}}$
(6) $\lambda_{eff}=\lambda_{l}(1+\frac{\xi_{0}}{r})^{0.5}$ (7)

$\xi_{0}=\frac{0.18\hbar v_{f}}{k_{B}T_{C}}$ is the intrinsic coherence
length, $v_{F}$ Fermi velocity, $\lambda_{l}$ London penetration depth and
$\bf\,r$ is the average size of the finite size superconducting clusters.
According to Eqn. 5 and Eqn. 6 in the finite size regime the values of
critical parameters are decided by the size of superconducting volume
fraction, hence larger amplitude of $H_{C2}$ (and JC) in case of A3 compared
to A2 and A4 indicates the superconducting volume fraction is smaller in A3
compared to A2 and A4. According to Eqn. 7 the effective London penetration
depth also increases in the finite size region, which causes the reduction of
effective diamagnetic fraction (or Meissner fraction) and hence smaller value
of superconducting volume fraction have been observed in case of A3 compared
to A4 and A5 (as shown in FIG. 4(b)). XRD results are already depicting that
there is no change of crystalline volume of LCu in any of the composite
compared to parent LCu. So, decrease of the superconducting volume fraction
and increase of $H_{C2}$ and JC is not due to reduction of crystalline size.
Therefore, the SPM type AFM clusters which are formed within the bulk of LCu
reduces the superconducting volume fraction, as the microscopic spin structure
of the corresponding SPM cluster are static in nature and can not support
superconductivity. It is already known from previous studies that any kind of
static magnetic structure is unfavorable for superconductivity in cuprate
superconductors, because the superconducting state of a cuprate evolves after
diluting the antiferromagnetic network and only the dynamic AFM fluctuation
can sustain with superconducting ordering [1, 2, 3]. Therefore, these
observations are suggesting that the whole superconducting volume is divided
into two phase separated regions, one phase consists of isolated SPM type AFM
clusters and another phase is the finite size superconducting clusters.
So, from all the above discussion it can be claimed that when AFM fluctuation
is stabilized by the exchange magnetic field of LCo, the AFM order emerges and
superconducting volume fraction suppresses. This clearly indicates that the
AFM type fluctuation can be a mediator of superconductivity in cuprate
superconductor indicating that it is taking part in Cooper pairing.

##  V Conclusion

Composite of a superconductor (LCu) and a ferromagnet (LCo) has been prepared
by solid state reaction method. In ac-susceptibility measurement an anomaly is
observed above $T_{S(onset)}$ and it has been proved as the interface effect
by changing the effective interface between LCu and LCo. It is proved through
the studies of the linear ($\chi_{1}$) and nonlinear ac-susceptibilities
($\chi_{2}$, $\chi_{3}$) that the exchange field of LCo affects the copper
spins and stabilizes the short range ordered AFM state within the bulk of LCu.
These observations unambiguously indicates that the dynamic AFM fluctuation
can be the mediator of copper pairing in cuprate superconductor substantiated
by the observation of reduction of superconducting volume fraction when the
AFM fluctuations are stabilized by formation of stable short range ordered SPM
type AFM phase within the bulk. Further, the superconducting volume fraction
reduce to the quantum size limit and two phase separated regions are formed,
one is super-paramagnetic type AFM clusters and the rest is finite size
superconducting volume. Another important inference is that, the reduced
superconducting volume fraction shows quantum size effect (According to
DeGennes and Tinkham model), which is otherwise very difficult to achieve by
preparing the nanoparticle of LCu, because of the structural limitation and
degradation of oxygen stoichiometry. So these composites provide an unique
path to understand the mechanism of Cooper pairing and also opens up a pathway
to study the physical property of the finite size superconducting clusters of
cuprate superconductors.

##  VI Acknowledgment

We are thankful to Dr. Mukul Gupta and Dr.N.P lalla for XRD measurements.
Kranti Kumar Sharma and Dr.Santanu De are acknowledged for discussion and help
during measurements. We are thankful to IRCC IIT Bombay MEMS department for
TEM measurement.

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