# Observation of an unconventional giant negative exchange bias effect in
La0.5Sr0.5Co0.85Nb0.15O3

Rishabh Shukla  Department of Physics, Indian Institute of Technology Delhi,
Hauz Khas, New Delhi-110016, India  B. Schwarz  Institute for Applied
Materials (IAM), Karlsruhe Institute of Technology (KIT), 76344, Eggenstein-
Leopoldshafen, Germany  R. S. Dhaka<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016,
India

###### Abstract

We find an unconventional giant negative exchange bias (EB) of $H_{\rm EB}$ =
–14.1 kOe at 2 K (cooling field of 50 kOe) in the cluster spin-glass (CSG)
La0.5Sr0.5Co0.85Nb0.15O3 perovsikte cobaltites. The magnetic memory effect,
aging measurements, and nonlineraity in specific heat capacity reveal the
glassy magnetic state at low temperatures. Further, the detailed analysis of
ac-magnetic susceptibility confirms the glassy state below $\sim$58 K and the
obtained characteristic spin-relaxation time-scale of $\tau_{0}$ =
8.4$\times$10-10 s indicates the presence of CSG. Moreover, the analysis of
magnetic training effect using the classical EB relaxation model reveals that
the frozen spins relax slowly as compared to the rotatable spins at the
interface of antiferromagnetic/ferromagnetic (AFM/FM) regions in CSG.
Interestingly, the dependence of EB parameters is found to be unconventional
for cooling field $>$50 kOe as the $H\rm_{EB}$ and $M\rm_{EB}$ show decreasing
trend instead of expected saturation at higher fields. This unusual nature
emerges due to large negative values of intrinsic interface exchange coupling
($J_{i}$), i.e., –10.24$\pm$0.22 meV and –12.55$\pm$0.49 meV for the measuring
fields of $\pm$50 kOe and $\pm$90 kOe, respectively, whereas the number of
spins in the FM cluster ($N_{\rm FM}$) are found to be small in the range of
2.4–3.1. These obtained values of $J_{i}$ and $N_{\rm FM}$ indicate the
dominant AFM interactions and the presence of FM clusters in the AFM matrix,
respectively, which correlate well with the observed unconventional behavior
of giant negative exchange bias in the present sample.

##  I  Introduction

The essence of futuristic magnetic materials having large values of saturation
magnetization and coercive fields have been demonstrated through the
artificially developed exchange bias (EB) mechanism [1] for the magnetic
storage devices, spin-valves, high-density magnetic recordings, domain
stabilizers, permanent magnets, etc. [2, 3, 4, 5]. The EB phenomenon was first
discovered in the magnetic nanostructures of Co nanoparticles coated with a
layer of CoO [1]. Further, the EB effect is observed in the thin-film
heterostructures via coupling of ferromagnetic (FM) and antiferromagnetic
(AFM) layers, inhomogeneous spin-glasses via coupling of FM and AFM domains,
artificial FM/AFM superlattices, etc. [8, 9, 6, 7]. In general, the EB effect
is understood in terms of the interface exchange coupling between FM and AFM
layers, which leads to the unidirectional exchange anisotropy, and a shift in
the isothermal magnetization (M–H) loops is observed on cooling the sample
below the Néel temperature in the presence of an applied magnetic field [1, 6,
11, 10]. Moreover, this understanding was further extended to the EB effect
observed at the interface of FM–spin glass (SG), FM–ferrimagnet (FIM), AFM–SG,
AFM–FIM–SG, as well as solely in cluster spin glass (CSG) system [12, 13, 14].
However, the magnitude and characteristics of the observed EB effect are
greatly influenced by the choice of interface, type of magnetic interactions,
thickness of the interface, etc. [3, 11]. For example, in the AFM–FM interface
(since FM and AFM have small and large anisotropy, respectively), on cooling
the sample below TN the spins orient towards the applied magnetic field;
however, at the interface, the spins exhibit a unidirectional anisotropy and a
large enough measuring field overcomes the interface magnetization and
reverses the FM [3]. These can result in shifting the hysteresis loop by the
magnitude of the interlayer exchange coupling opposite to the applied magnetic
field [15, 16]. This suggests that an interface is required to induce the
pinning effect for the observation of the EB. This pinning effect need not
originate because of the uncompensated spins, but it can also emerge due to
the presence of frozen spins, very likely in the glassy phases [17].

Note that, for effective utilization of the EB effect, a comprehensive
understanding of the EB effect in terms of interface magnetism, disorder,
inhomogeneous magnetization, etc. is vital and plays a crucial role in
designing the materials of desirable applications [18, 6, 11, 19]. Moreover,
the use of SG as an essential ingredient in these artificial EB systems is
very useful to widen the understanding of the effects of frozen spins,
frustration, and inhomogeneous magnetic regions. In this line, the SG-
supported EB systems are also technologically useful due to their presence in
magnetically disordered materials and their composites [20, 21]. In order to
explain the SG phenomenon, the droplet model [22, 23] and hierarchical models
[24] are used, which utilize the multi-valley energy structure and equivalent
spin configurations. Despite several efforts to understand the EB effect in
single-phase magnetically inhomogeneous compounds [12, 25] the understanding
of the exchange bias phenomenon in SG phases is elusive. Therefore, a complete
picture of the spin-glass driven EB is vital for engineering the spintronics
devices, magnetic memory devices, and other technological advances [7, 26]. In
transition metal oxide the EB has been investigated under different schemes,
which involve, thin film interfaces, core-shell nanoparticles, cationic
substitution in magnetically ordered systems, etc. [27, 28, 29, 30]. However,
the observation of EB and its origin in polycrystalline samples remain
debatable. For example, in cobaltites, the observed magnitude of EB varies
significantly, i.e., 5.5 kOe in SrLaCo0.5Mn0.5O4 [28], 1.4 kOe in
Sr1.5Pr0.5CoO4 [31], 1.1 kOe in La1.5Ca0.5CoIrO6 [29], 1.2 kOe in LaSrCoFeO6
[32], 2.1 kOe in La1.5Ca0.5CoFeO6 [33], 5.1 kOe in Sm1.5Ca0.5CoMnO6 [34], 6
kOe and 9.5 kOe in SrLaFe0.25+xMn0.25Co0.5-xO4 [35], 8.22 kOe in SrCo1-xVxO3-δ
[30], 0.8 kOe in Sr2CoNbO6 [18], etc.

Furthermore, the hole doping of divalent cations in LaCoO3, i.e., La1-xSrxCoO3
enthralled the researchers due to the observed colossal magnetoresistance
(CMR), which is believed to have originated from the presence of magnetic
nonhomogeneity in the samples [36, 37, 38]. The substitution of diamagnetic
Nb5+ (4$d^{0}$) in La0.5Sr0.5Co1-xNbxO3 converts the Co4+ ions into Co3+ ions,
and therefore an increase in the density of Co3+ over Co4+ ions [39, 40]. The
FM clusters appear due to the double exchange (DE) interactions from the
Co4+-O-Co3+, whereas the AFM clusters emerge due to the super-exchange (SE)
interactions from Co3+-O-Co3+ and Co4+-O-Co4+ [41, 42]. The electrons in the
outer shell of transition metals are localized, and hence the interactions
strongly depend on the orientation of the localized spin moments [41, 43].
Also, the observed magneto-electronic intrinsic phase separation (MIPS) plays
a crucial role in the generation of the EB in these complex oxides owing to
the presence of hole-rich metallic FM clusters and hole-poor insulating AFM
clusters [44, 45].

Therefore, we investigate the evolution of exchange bias (EB), memory effect
and heat capacity in bulk La0.5Sr0.5Co0.85Nb0.15O3 sample. The ac-magnetic
susceptibility data confirms the glassy nature of the sample below $\sim$58 K
(Tf) and the estimated characteristic time-scale of $\tau_{0}$ =
8.4$\times$10-10 s through detailed analysis suggests that the individual spin
entities behave like the cluster spin-glass (CSG). Moreover, the nonlinearity
in specific heat capacity at low-temperature is analyzed including the
magnetic contributions and further validates the glassy magnetic state.
Intriguingly, we find an unconventional behavior of giant negative exchange
bias -14.1 kOe (at 2 K) and -7.4 kOe (at 5 K) for a 50 kOe cooling field. The
exchange interactions in CSG at the interface of FM and AFM clusters
originated due to the hole-rich and hole-poor regions, respectively. The
exchange bias parameters (HEB and MEB) decrease at the higher cooling fields
($>$50 kOe) which are modeled with the intrinsic interface exchange coupling
and reveals $J_{i}=$ -10.24$\pm$0.22 meV and $N_{FM}=$ 2.4$\pm$0.2 (measuring
field of $\pm$50 kOe) and $J_{i}=$ -12.55$\pm$0.49 meV and $N_{FM}=$
3.1$\pm$0.1 (measuring field of $\pm$90 kOe), which is responsible for the
reduction in the EB parameters at higher cooling fields.

##  II  Experimental Details

Polycrystalline sample La0.5Sr0.5Co0.85Nb0.15O3 was synthesized using the
solid-state reaction route, where stoichiometric amount of as purchased SrCO3,
Co3O4, Nb2O5, and La2O3 (dried at 900oC for 6 hrs) were mixed homogeneously
using an agate mortar-pestle and then heated in air using a muffle furnace
(from Nabertherm) at 1000oC (48 hrs) and 1300oC (36 hrs) with intermittent
grindings. The basic characterizations including structural, magnetization,
transport and electronic properties of this sample are reported in [39]. The
detailed magnetic and specific heat measurements in different protocol are
performed using the DynaCool Physical Property Measurement System from Quantum
Design, USA at Karlsruhe Institute of Technology (KIT), Germany.

##  III  Results and Discussion

In Fig. 1(a) we show the temperature-dependent dc-magnetic susceptibility
($\chi\rm_{dc}$) of La0.5Sr0.5Co0.85Nb0.15O3 measured at 10 kOe in both zero-
field-cooled (ZFC) and field-cooled (FC) protocols. The $\chi\rm_{dc}$
exhibits a monotonous increase with lowering the temperature till $\approx$47
K (termed as freezing temperature, Tf); however, below Tf, the $\chi\rm_{dc}$
decreases in the ZFC mode, while increases at a slower rate in the FC mode. A
clear bifurcation/separation in the ZFC-FC curves near 53 K (termed as
irreversibility temperature, Tirr) and the observed cusp/peak in the ZFC curve
are resultant of the competition between the energy of anisotropic field [due
to the ferromagnetic (FM)/antiferromagnetic (AFM) interactions] and the
applied field energy [46], which is considered to be a favorable condition for
the spin glass (SG) behavior [47, 48] and exchange bias [12]. Also, the
observed shift in the Tf value from 55 K at 100 Oe [39] to 47 K at 10 kOe (see
inset) advocates for the presence of SG state below the Tf in the sample. To
understand further, we perform the memory effect measurements under the ZFC
and FC protocols where initially the sample is cooled from 300 K to 2 K with a
cooling rate of 2 K/min in the absence of the magnetic field without any halts
and then measured the magnetization with the conventional method (during
warming) at an applied field of 100 Oe with a heating rate of 2 K/min up to
300 K [this ZFC curve is marked as reference curve (solid red line) in Fig.
1(b)]. Now again the sample was cooled from 300 K to 2 K in the absence of a
magnetic field at 2 K/min rate, but with the intermediate halts at 100 K
($>$Tf) and 20 K ($<$Tf) for 2 hrs each, and then measured the magnetization
during warming at 100 Oe [this ZFC curve is marked as memory curve (solid
black line) in Fig. 1(b)]. A shallow dip is observed for the memory curve near
the 20 K halt temperature (below Tf) even in the absence of a magnetic field;
however, no difference in the curves near the 100 K halt temperature (above
Tf). This dip in the ZFC memory curve with respect to the ZFC reference curve
is highlighted in the inset of Fig. 1(b).

Figure 1: (a) The zero-field cooled (ZFC) and field cooled (FC) magnetization
curve recorded at 10 kOe. The insets show the zoomed view near (a1)
T$\rm{}_{f}=$ 47 K and (a2) T$\rm{}_{irr}=$ 53 K. The magnetic memory effect
in magnetization under (b) ZFC and (c) FC protocols at 100 Oe. Note that halts
at 20 K and 100 K were employed during the measurement for 2 hrs each. The
inset in (b) highlights the memory effect under ZFC protocol, and inset in (c)
highlights the magnetization drop in FCW at 20 K ($<$Tf) and no change at 100
K ($>$Tf).

Moreover, the difference curve between the reference and memory ZFC curves
exhibit a dip near 20 K only, which confirms that the system keeps the
sample’s history and indicates the glassy nature below the Tf. Further, we
measure the memory effect under the FC protocol, where initially sample was
cooled from 300 K to 2 K with a magnetic field of 100 Oe at 2 K min-1 rate
without any halts, and measured the magnetization during warming with a
heating rate of 2 K min-1 [this FCW curve is marked as reference curve (open
red circles) in Fig. 1(c)]. Then, the sample was cooled from 300 K to 2 K with
a magnetic field of 100 Oe at a rate of 2 K min-1 including the intermediate
halts at 20 K ($<$Tf), and 100 K ($>$Tf) for 2 hrs each. Note that, the
magnetic field is switched off during the halt period and then switched on
during the subsequent sample cooling. In this case, the magnetization is
measured during sample cooling and warming, and these curves are termed as FCC
(open blue triangles) and FCW (solid green circles) in Fig. 1(c),
respectively. The memory effect prevails below Tf where the magnetization
decreases near 20 K, while above Tf there is no difference around 100 K, see
the inset for clarity. These above observations suggest the presence of
competitive/coupled FM–AFM interactions in the sample resulting the glassy
magnetic state.

Now, first we investigate the spin dynamics of the glassy magnetic state
through detailed analysis of the ac-magnetic susceptibility ($\chi\rm_{ac}$)
data in terms of the real ($\chi$’) and imaginary ($\chi$”) parts, as shown in
Figs. S1(a, c) and Figs. S1(b, d) of [49], respectively. The peak value in
$\chi$’ is found to be 58.2 K at 193 Hz excitation frequency, which shifted to
60 K at 9984 Hz along with the decrease in the magnitude of the magnetic
susceptibility. Here, we first plot the variation of peak temperature with the
excitation frequency (not shown) and extract the Mydosh parameter using the
formula [50]: $\delta{T_{f}}=(\Delta{T_{f}})/[T_{f}\Delta(log_{10}f)]$ where
Tf is the freezing temperature and $f$ is the excitation frequency. The
obtained value of $\delta{T_{f}}=$ 0.017 is found to be higher as compared to
the canonical spin glasses (AuMn, $\delta{T_{f}}$ = $\sim$0.0045 [51] and
CuMn, $\delta{T_{f}}$ = $\sim$0.004 [50]) and lower as compared to the
superparamagnets ($\alpha$-Ho2O3(B2O3), $\delta{T_{f}}$ = $\sim$0.28 [50]).
However, we note that the value for the present sample lies within the range
of the cluster spin glasses (0.01–0.018) [52, 53]. Also, the obtained values
of characteristic spin relaxation time from the analysis of aging measurements
found to be consistent with glassy systems [54, 55], as shown in Fig. S3 and
Table S1 of [49].

Figure 2: (a) The dependence of the spin relaxation time ($\tau$) on the
freezing temperature (Tf) with an exponential fit (solid black line) using the
dynamical scaling power law, and (b) the dependence of freezing temperature Tf
on 1/ln($\tau/\tau_{0}$) with a linear fit (solid black line) following the
Vogel-Fulcher law.

Further, to extract the time scale of spin relaxation and type of interaction
between magnetic clusters, the frequency dependence of the Tf can be further
analyzed with the dynamic scaling theory using the power law [56],

$\displaystyle\tau=\tau_{0}\left(\frac{T_{F}-T_{SG}}{T_{SG}}\right)^{-z\nu^{\prime}}$
(1)

where $\tau$ is the relaxation time corresponding to the measured frequency
($f=$ 1/$\tau$), $\tau_{0}$ is the characteristic relaxation time for the
flipping of a single spin entity, TSG is the spin-glass temperature or termed
as the freezing temperature when $\tau$ diverges or frequency tends to zero,
and z$\nu^{\prime}$ is the dynamic critical exponent [where critical exponent,
$\nu^{\prime}$ is associated with the correlation length $\xi$ =
(Tf/TSG-1)${}^{\nu^{\prime}}$ and dynamical critical exponent, $z$ relates
correlation length to spin relaxation time such that $\tau\sim\xi^{z}$]. The
best fit (solid black line) to the experimental data using equation 1, as
shown in Fig. 2(a), yields the values of $\tau_{0}$ = 8.4$\pm$0.3$\times$10-10
s, TSG = 56.5$\pm$0.1 K, and z$\nu^{\prime}$ = 4.5$\pm$0.4. For the canonical
and cluster spin glasses, the $\tau_{0}$ exists in the range of 10-12 to 10-14
s and 10-6 s to 10-10 s, respectively, and z$\nu^{\prime}$ lies between 4 and
12 [53, 57]. In the present case, the obtained value of $\tau_{0}$ confirms
the presence of cluster spin glass and indicates the slower spin dynamics.
Also, we investigate the spin-interaction by the Arrhenius relation, as shown
in Fig. S2(a) of [49], which manifests the presence of interaction between the
individual spins (cluster spin glass). Moreover, the dynamical scaling
behavior of spin-freezing is modified to the Vogel-Fulcher law, as given below
[50, 53, 56],

$\displaystyle\tau=\tau_{0}exp\left(\frac{-E_{a}}{k_{B}(T_{f}-T_{0})}\right)$
(2)

where T0 is the characteristic temperature/Vogel-Fulcher temperature utilized
to measure the strength of interaction between magnetic entities and other
parameters are same as defined earlier. A simplified form of Vogel-Fulcher law
is used to plot a graph between Tf versus 1/ln($\tau/\tau_{0}$) in Fig. 2(b)
and a best fit (solid black line) to the experimental data using equation 2
results in the values of Ea = 6.8$\pm$1 meV, and T0=53.0$\pm$0.5 K. Here, a
non-zero value of T0 (comparable to the freezing temperature, Tf) in the
present sample further validate the interaction between the magnetic clusters
[57].

Figure 3: The isothermal M–H loops recorded under the (a) ZFC protocol (b) FC
protocol with a cooling field of 70 kOe and within $\pm$70 kOe, (c) with a
cooling field of 90 kOe and within $\pm$90 kOe, (d) with a cooling field of 50
kOe and within $\pm$50 kOe, (e) with a cooling field of 50 kOe and within
$\pm$90 kOe, and (f) a comparison of the M–H loops measured at different
cooling fields and between different measuring fields at 5 K; insets in (a–c)
highlight the zoomed view near origin.

It is interesting to note here that the competitive AFM-FM interactions
resulting in the cluster spin glass behavior can give rise to the exchange
bias (EB) effect in the sample. Therefore, in order to get a complete picture
of FM-AFM interactions and to investigate the EB effect we perform the
isothermal field-dependent magnetization (M–H) below the Tf under different
cooling protocols. In this line, first, we recorded the isothermal M–H loops
under the ZFC protocol with the maximum applied magnetic field of $\pm$90 kOe
at 5 K, 10 K, 30 K, and 50 K and a comparison of these curves is presented in
Fig. 3(a) where the inset showed the zoomed view within $\pm$6 kOe. Here, we
define the coercivity $\rm H_{C}=\rm(|H_{C_{1}}|+|H_{C_{2}}|)/2$, where
H$\rm{}_{C_{1}}$ and H$\rm{}_{C_{2}}$ are the left (-ve H-axis) and right (+ve
H-axis) coercive fields; and retentivity $\rm
M_{r}=\rm(|M_{r_{1}}|+|M_{r_{2}}|)/2$, where M$\rm{}_{r_{1}}$ and
M$\rm{}_{r_{2}}$ are the top (+ve M-axis) and bottom (-ve M-axis) magnetic
retentivity values. The estimated values of HC and Mr are found to be 5.2 kOe
and 129 emu mole-1 at 5 K; 4.2 kOe and 104 emu mole-1 at 10 K; 1.8 kOe and 56
emu mole-1 at 30 K; and 84 Oe and 3 emu mole-1 at 50 K, respectively.
Interestingly, the values of HC and Mr increase monotonously with a decrease
in the temperature for T$<$Tf, which indicate the dominance of FM interactions
at low temperatures. Further, to elucidate the phenomenon of exchange bias we
measure the isothermal M–H loops under the FC protocol in the cooling field of
$\pm$70 kOe at 5 K, 10 K, 30 K, and 50 K, as shown in Figs. S4(a–d) of [49],
respectively. These results for $\leq$ 30 K manifests the shift in the M–H
loop in opposite direction of the cooling field, but equal magnitude; however,
no shift is observed at 50 K (see Fig. S4 of [49]). This observed exchange
bias effect below Tf affirms the intrinsic coupling between the FM–AFM
interactions present in the sample. Another important observation is that
there is no opening in the M–H loops in the high-field region when recorded
under the ZFC protocol at different temperatures [see Fig. 3(a)]; however, in
case of the FC protocol we find an opening in the M–H loops at high-field
region and have the correlation with the direction of the cooling field [see
Fig. S4 of [49]]. This opening in the M–H loops can be ascribed due to the
time-dependent variation of the magnetization at high applied magnetic field.
Further, we show the isothermal M–H loops under the FC protocol at various
temperatures in Figs. 3(b) for the cooling field of 70 kOe (measuring field of
$\pm$70 kOe) and in Figs. 3(c) for the cooling field of 90 kOe (measuring
field of $\pm$90 kOe) where the respective insets compare the zoomed view near
origin. Moreover, we can estimate the exchange bias field (HEB) and exchange
bias coercivity (MEB) from the isothermal M–H loops recorded under the FC
protocol using $\rm H_{EB}=\rm(H_{C_{1}}+H_{C_{2}})/2$ and $\rm
M_{EB}=\rm(M_{r_{1}}+M_{r_{2}})/2$. We compared these parameters for different
cooling fields at few temperatures, the HC and Mr in Fig. S5(a) of [49], and
the HEB and MEB in Fig. S5(b) of [49]. We find that the HC and Mr show nearly
a linear decrease; while the HEB and MEB exhibit a logarithmic decrease as an
increase in temperature affects the alignment of the spins significantly even
below the Tf and almost randomizes the spins above the Tf (discussed in detail
later).

Moreover, the isothermal M–H loops are measured under the FC protocol with a
constant cooling field of 50 kOe between 2–50 K, by varying the measuring
fields within $\pm$50 kOe and $\pm$90 kOe, as shown in Figs. 3(d, e),
respectively. First we find a significant effect of measuring/cooling field;
therefore, a detailed comparison of the M–H loops is shown in Figs. S6(a–f) of
[49] at various temperatures. Since the measuring field can be understood as a
force to align the frozen/blocked spins along the field direction, and
therefore, we observe a lower reduction of exchange bias parameters at higher
temperatures as compared to low temperatures. In Fig. S6(a) of [49], we find
that the M–H loop measured at 2 K shifted completely to the -ve H-axis and +ve
M-axis, which manifests that $\pm$50 kOe magnetic field is not sufficient to
realign the frozen spins, while a measuring field of $\pm$90 kOe seems to be
enough to realign the frozen spins in the field direction (see Fig. S6(a) of
[49]). This indicates that the measuring field range also plays an important
role in the behavior of the exchange bias effect. The values of HEB (MEB) and
HC (Mr) as highlighted in Figs. S6(a) of [49] are estimated via the average
value of the field where curves intercept the H-axis (M-axis) and half-width
of the loop where curve intercepts the M-axis (H-axis), respectively. These
extracted exchange bias parameters are summarized in Table S2 of [49], where
we observe very large values of HEB = -14.1 kOe and MEB = 359.6 emu mole-1,
which are found to be highest reported for the perovskite cobaltites in the
literature as per the best of our knowledge. The values of HEB and MEB
decreases to -8.23 kOe and 206.4 emu mole-1, respectively, for the measuring
field of $\pm$90 kOe. We also find decrease in the values of these parameters
with increasing the temperature. Further, to understand the exchange bias
behavior we measure the M–H loops at 5 K at different measuring and cooling
fields in such a way that we first cool the sample in a constant magnetic
field and then the M–H loops are measured within the $\pm$cooling field, as
shown in Fig. 3(f). In order to highlight this effect, we compare the
isothermal M–H loops in Figs. S7(a–f) of [49] measured at various constant
cooling fields, while varying the measuring field range. Here, while lowering
the measuring field, we observe that $\leq$50 kOe cooling field is not
sufficient to realign the frozen spins even at 5 K temperature and therefore
the M–H loops are shifted towards the -ve H-axis and +ve M-axis. However, a
measurement field of up to $\pm$90 kOe is sufficient to realign the frozen
spins. The parameters obtained from these M–H loops are summarised in Table S3
of [49].

Figure 4: (a) The isothermal M–H loops recorded up to consecutive eight ($n=$
1–8) cycles within $\pm$90 kOe at 5 K under the FC protocol with a cooling
field of 50 kOe, an enlarged view of the decay in the (b) HC1 (-ve branch) and
(c) HC1 (+ve branch), the variation in the (d) HC1 (-ve branch) on the left-
axis and HC2 (+ve branch) on the right-axis, and (e) the Mr1 (+ve branch) on
the left-axis and Mr2 (-ve branch) on the right-axis with the number of
subsequent cycles.

In order to highlight the observation of an unconventional exchange bias
behavior we measure the isothermal M–H loops at 5 K within $\pm$90 kOe
measuring field by varying the cooling field in the range of 10–90 kOe, as
shown in Fig. S8(a) of [49]. The zoomed view in Fig. S8(b) of [49] highlights
the variation of the +ve and -ve branches of the coercive field, where we can
clearly see that the +ve branch of coercive field exhibits a smaller variation
with the cooling field while the -ve branch of coercive field changes
significantly. We observe that for the cooling fields up to 50 kOe, the value
of the coercive field increases, i.e., the -ve branch of the M-H loop
intercepts the H-axis increases (marked with the arrow); while, for the
cooling fields beyond 50 kOe, a decrease in the value of intercept on the -ve
H-axis is observed up to a cooling field of 90 kOe, as highlighted by arrows
in Fig. S8(c) of [49]. At large cooling field, the size of the FM cluster
grows, and therefore a smaller measuring field is enough to realign the frozen
spins which results in a decrease in the exchange bias. The extracted
parameters are summarized in Table S4 of [49].

It is important to study the magnetic training effect (MTE), which helps to
understand the intrinsic exchange bias where the values of HC and HEB are
expected to exhibit a dramatic reduction as the sample is continuously field
cycled at a fixed temperature below Tf [58, 59]. Therefore, eight consecutive
isothermal M–H loops were measured at 5 K after a FC protocol with a cooling
field of 50 kOe, as shown in Fig. 4(a). Further, we highlight the shift in the
M–H loop branches on the -ve and +ve H-axis in Figs. 4(b, c), respectively. We
find an abrupt drop in the values of HC and HEB from cycle 1 to cycle 2, and
it further decreases when the M–H loops are measured in the consecutive
cycling. The MTE help to understand the effect of magnetization due to the FM
and AFM interface in the sample and a decrease in the coercive fields (HC) and
remanence magnetization (Mr) with the consecutive M–H loops manifest that
frozen spins relax at the interface [60]. We plot the values of the HC1 (-ve
branch) on the left-axis (solid red circles) and HC2 (+ve branch) on the
right-axis (open blue circles) in Fig. 4(d), and Mr1 (+ve branch) on the left-
axis (solid purple triangles) and Mr2 (-ve branch) on the right-axis (open
green triangles) in Fig. 4(e) as a function of number of cycles/loop index
($n$). We find that the values on -ve H-axis (HC1) change from -16.2 kOe ($n=$
1) to -12.5 kOe ($n=$ 8); whereas on the +ve H-axis (HC2), the change is from
5.9 kOe ($n=$ 1) to 6.8 kOe ($n=$ 8), see Fig. 4(d). Like wise, the values of
Mr1 changes from 450 emu/mole ($n=$ 1) to 300 emu/mole ($n=$ 8); whereas the
Mr2 changes from -180 emu/mole ($n=$ 1) to -194 emu/mole ($n=$ 8), see Fig.
4(e).

Moreover, to understand the MTE quantitatively we follow the idea of Binek
$et~{}al.$ that the training of antiferromagnetic spins at the FM–AFM
interface gives rise to the reorientation of the domains, i.e., $n$ dependence
of exchange bias parameters (HEB/MEB) [61]. Therefore, a power law fit is used
to approximate the arrangement of spins at the interface and can be simulated
by the following equations for $n>$ 1 [61],

$\displaystyle\rm H^{n}_{EB}-H^{\infty}_{EB}$ $\displaystyle=$
$\displaystyle\frac{k_{H}}{\sqrt{n}}~{}~{}~{}~{}\rm and~{}~{}~{}~{}\rm
M^{n}_{EB}-M^{\infty}_{EB}=\it\frac{k_{M}}{\sqrt{n}}$ (3)

where H$\rm{}^{n}_{EB}$ (M$\rm{}^{n}_{EB}$) and H$\rm{}^{\infty}_{EB}$
(M$\rm{}^{\infty}_{EB}$) are the EB fields (magnetization) in the nth cycle
and the limit of infinite loops, respectively; and kH, kM are the system-
dependent constants corresponding to the EB field and magnetization,
respectively; and $n$ is the number of cycles. The best fits to the calculated
-HEB (solid black line) and MEB (solid black line) using the power law
equations 3 for $n>$ 1 are presented in Fig. 5(a) and the obtained fitting
parameters are summarized in Table S4 of [49]. Note that the extrapolation of
the power law fit up to $n=$ 1 underestimates the values of HEB (-4.84 kOe as
compared to -5.15 kOe) and MEB (128.6 emu mole-1 as compared to 135.6 emu
mole-1). This difference in the values is correlated to the abrupt drop in the
-HEB and MEB values from $n=$ 1 to $n=$ 2, while for $n$$\geq$2 the drop is
monotonous and therefore follow the power law precisely to the experimental
data for $n>$ 1\. This behavior is correlated to the spin relaxation at the
FM/AFM interface while field-switching and a uniform decrease is related to
the thermally activated spin-rearrangement at the magnetically disordered
FM/AFM interface [62].

Figure 5: (a) The variation of exchange bias field (HEB, open red circles on
left-axis) and magnetization (MEB, open green triangles on right-axis) with
the number of cycles ($n$). The power-law fit (solid black lines) to HEB and
MEB for $n>$1 using the equations 3, respectively, and the dotted black line
extrapolates the same up to $n=$1, (b) the experimental (open symbols) and
theoretically generated (solid symbols) values of HEB and MEB using equations
4 and 5, respectively, (c) the best fits (solid black lines) to the dependence
of HEB and MEB with $n$ using equations 8 and 9, respectively, and (d) the
relative reduction of the HEB and MEB values with $n$ according to the
equations 10 and 11, respectively.

To get the exchange bias, the large AFM spin correlation is necessary to pin
the FM layer in the process of magnetization reversal. The relaxation of the
spins to the equilibrium because of the applied magnetic field scales to the
third power of order parameter and therefore, the non-exponential relaxation
of the FM spins below Tf can be approximated with the equations given below
[61];

$\displaystyle\rm H_{EB}(n+1)$ $\displaystyle=$ $\displaystyle\rm
H_{EB}(n)-\gamma^{\prime}_{H}\left[(H_{EB}(n)-H^{\infty}_{EB})\right]^{3}$ (4)
$\displaystyle\rm M_{EB}(n+1)$ $\displaystyle=$ $\displaystyle\rm
M_{EB}(n)-\gamma^{\prime}_{M}\left[(M_{EB}(n)-M^{\infty}_{EB})\right]^{3}$ (5)

where HEB($n$) and MEB($n$); HEB($n+$1) and MEB($n+$1)); and
H$\rm{}^{\infty}_{EB}$ and H$\rm{}^{\infty}_{EB}$ are the EB fields and
magnetization in the nth, (n+1)th cycle and in the limit of infinite loops,
respectively. Here, the $\gamma^{\prime}_{H}$ and $\gamma^{\prime}_{M}$ are
the system dependent constant and can be calculated from the experimental data
using the following recursive relations [61],

$\displaystyle\rm\gamma^{\prime}_{H}=\frac{1}{N-1}\sum_{n=2}^{N}\frac{\left[H_{EB}(n)-H_{EB}(n+1)\right]}{\left[H_{EB}(n)-H_{EB}^{\infty}\right]^{3}}$
(6)
$\displaystyle\rm\gamma^{\prime}_{M}=\frac{1}{N-1}\sum_{n=2}^{N}\frac{\left[M_{EB}(n)-M_{EB}(n+1)\right]}{\left[M_{EB}(n)-M_{EB}^{\infty}\right]^{3}}$
(7)

where the value of H$\rm{}_{EB}^{\infty}$ (M$\rm{}_{EB}^{\infty}$) is utilized
as additional input from the power-law fit [equation 3]. The optimized values
of $\gamma^{\prime}_{H}$ and $\gamma^{\prime}_{M}$ according to the equation 3
are included in Table S4 of [49]. The HEB and MEB values are theoretically
generated for $n>$ 1 using the above equations and superimposed to the
experimental data in Fig. 5(b), where we find that on the left-axis (right-
axis), experimentally and theoretically generated HEB (MEB) are shown in open
red circles (open green triangles) and closed blue circles (closed black
triangles), respectively. The generated values of HEB and MEB are in good
agreement with the experimental data except for a small deviation for the
higher values of $n$.

Since the presence of spin frustration at the FM/AFM interface modify the
magnetic anisotropy leads to the two different species of uncompensated spins,
termed as the frozen and rotatable spins [62], which are strongly exchange
coupled to the AFM and FM layers, respectively. This idea of two types of
uncompensated spin species can be understood by a quantitative model proposed
for the relaxation/decay of the exchange bias with number of cycles where the
EB field can be estimated using the equations below [62],

$\displaystyle\rm H^{n}_{EB}=\rm
H^{\infty}_{EB}+A_{f}exp\left[\frac{-n}{P_{f}}\right]+A_{r}exp\left[\frac{-n}{P_{r}}\right]$
(8) $\displaystyle\rm M^{n}_{EB}=\rm
M^{\infty}_{EB}+A_{f}exp\left[\frac{-n}{P_{f}}\right]+A_{r}exp\left[\frac{-n}{P_{r}}\right]$
(9)

where Af and Pf are the parameters related to the change in frozen spins,
while Ar and Pr are the evolving parameters related to the rotatable spins.
Here, the parameters Pf and Pr are dimensionless, while the Af and Ar have the
dimensions of magnetic field and magnetization, respectively. The best fits
(solid black line) using the equations 8 and 9 for the HEB (on the left-axis)
and MEB (on the right-axis), respectively, are presented in Fig. 5(c) and the
output parameters are summarized in Table S4 of [49]. Here, we find that the
major contribution in the exchange bias appears to be dominated by the
uncompensated spins which are rotating at the interface (rotatable spins) as
compared to the frozen spins at the FM/AFM interface. Since the Pr and Pf have
a ratio of nearly 13:1 (for HEB) and 14:1 (for MEB), which manifests that the
frozen spins relax slowly as compared to the rotatable spins, and therefore it
indicates that the rotatable spins governs the magnetic training effect
observed in the sample with the number of cycles.

Figure 6: The best exponential fits (solid black lines) to the exchange bias
(a, c) field (HEB), and (b, d) magnetization (MEB) in 2–50 K range and
comparing the effect of cooling fields (50 kOe, 70 kOe, 90kOe) as well as the
measuring field range within $\pm$50 kOe, $\pm$70 kOe and $\pm$90 kOe. The
cooling field dependence of the (e) HEB and (f) MEB at 5 K with the best fits
(solid black lines) to the data using equation 15, and the change in (g) HC
and (h) Mr values with the cooling field, measured at 5 K.

Furthermore, the relaxation/decay in the EB field (magnetization) with the
number of cycles can also be affected due to the spin rearrangement at the
interface. The relative percentage of the reduction/decay in the EB field
[TEH(%)] and the magnetization [TEM(%)] can be estimated using the equations
below [63],

$\displaystyle\rm TE_{H}(\%)$ $\displaystyle=$
$\displaystyle\rm\left[1-\frac{(H^{1}_{EB}-H^{n}_{EB})}{H^{1}_{EB}}\right]\times
100\%$ (10) $\displaystyle\rm TE_{M}(\%)$ $\displaystyle=$
$\displaystyle\rm\left[1-\frac{(M^{1}_{EB}-M^{n}_{EB})}{M^{1}_{EB}}\right]\times
100\%$ (11)

where, the H$\rm{}^{1}_{EB}$ (M$\rm{}^{1}_{EB}$) and H$\rm{}^{n}_{EB}$
(M$\rm{}^{n}_{EB}$) are the exchange bias field (magnetization) at first and
nth cycles. We plot the TEH(%) and TEM(%) in Fig. 5(d) on the left-axis (open
blue circles) and right-axis (open purple triangles) using equations 10 and
11, respectively. Here, we observe that the exchange bias field
(magnetization) decreases very sharply to around 77% in the initial two
cycles, and after that this decay rate reduces, which is consistent with the
reduction in the amplitude of spin-rearrangement happening with the number of
cycles.

Now, we perform quantitative analysis to understand the origin of the exchange
bias where we find above that there is a logarithmic decrease of HEB and MEB
in 2–50 K range for the cooling fields of 90 kOe, 70 kOe, and 50 kOe, see
Figs. 3(b, c, e). We model this variation in HEB and MEB using the following
exponential relations [64],

$\displaystyle\rm H_{EB}$ $\displaystyle=$ $\displaystyle\rm
H_{EB}^{0}~{}exp(-T/T_{0})$ (12) $\displaystyle\rm M_{EB}$ $\displaystyle=$
$\displaystyle\rm M_{EB}^{0}~{}exp(-T/T_{1})$ (13)

where the T0 (T1) and the H$\rm{}_{EB}^{0}$ (M$\rm{}_{EB}^{0}$) are constants
and the extrapolation of HEB (MEB) to the absolute zero temperature,
respectively. The best fits (solid black lines) to the experimental HEB and
MEB data for the cooling fields of 50 kOe and 90 kOe for the measuring fields
of $\pm$90 kOe are presented in Fig. 6(a, b), respectively, and the output
parameters are summarized in Table S6 of [49]. Interestingly, an exponential
dependence of the exchange bias parameters originates from the
competitive/coupled FM–AFM interactions in the sample where the SE and DE
interactions are responsible for the presence of AFM and FM interactions,
respectively [64, 65]. Also, we compare the exponential decay of the exchange
bias parameters HEB and MEB with the different cooling (50 kOe and 70 kOe) and
measuring ($\pm 50$ kOe, $\pm$90 kOe$,\pm$70 kOe) fields in Figs. 6(c, d),
respectively along with the exponential best fits (solid blue lines). The
estimated exchange bias parameters H$\rm{}_{EB}^{0}$ and M$\rm{}_{EB}^{0}$ are
found to decrease for lower values of the cooling fields, i.e., for 50 kOe
(12.4 kOe and 299 emu/mole) as compared to 90 kOe (16.35 kOe and 403 emu/mole)
for the fixed measuring field of $\pm$90 kOe. However, this effect is variable
for the different values of measuring fields, i.e., for a higher value of
measuring field $\pm$90 kOe, the values of H$\rm{}_{EB}^{0}$ and
M$\rm{}_{EB}^{0}$ decrease stronger as compared to $\pm$70 kOe and $\pm$50 kOe
(see Table S6 of [49]). Further, we present the dependence of the exchange
bias field (HEB) and magnetization (MEB) with the cooling fields (0–90 kOe) at
5 K in Figs. 6(e, f), respectively. Here, we find that the HEB and MEB
increases up to 50 kOe cooling field and then start decreasing monotonously.
This behavior is contrary to the conventional saturating behavior of HEB and
MEB with the cooling fields, as observed in [35, 66]. However, note that
similar unconventional EB behavior is also reported in the cluster spin glass
compounds [67, 68]. Therefore, this observation manifests that the exchange
bias appearing due to the cooling fields up to 50 kOe enhances competitive
FM–AFM interactions and then after $>$50 kOe starts suppressing it (discussed
below in detail). The decrease in the HEB and MEB is associated with the
increase in the values of HC and Mr, see Figs. 6(g, h), respectively, which
also suggests the growth of FM clusters and/or alignment of the FM clusters in
the magnetic field direction for the higher values of cooling fields. However,
this increase in HC and Mr is more prominent for the measuring field range of
$\pm$90 kOe as compared to $\pm$50 kOe, which also supports the hypothesis of
more growth/alignment of FM clusters in the larger measuring field range.

Notably the decrease in HEB and MEB values for the cooling field $>$50 kOe is
found to be invariant of measuring field ranges, i.e., $\pm$50 kOe and/or
$\pm$90 kOe [see Figs. 6(e, f)], which confirms the intrinsic nature of the
observed unconventional behavior. The HEB values can be influenced to some
extent by the increase/growth in the size of the FM cluster and/or alignment
of the degree of the FM cluster at particular cooling field. The cooling field
strongly affects the orientation of the frozen spins and governs the
competitive/coupled FM–AFM interaction at the interfaces in the CSG phase.
According to the Bean’s relation for the FM–AFM interface, the exchange bias
field is inversely proportional to the thickness of the FM layer (tFM), as per
the equation below [69, 14],

$\displaystyle\rm-H_{EB}=J_{ex}\frac{S_{AFM}S_{FM}}{\mu_{0}t_{FM}M_{FM}}$ (14)

where Jex is the exchange integral across the FM/AFM interface per unit area,
SFM and SAFM are the interface ferromagnet and antiferromagnet magnetization,
respectively, and MFM is the magnetization of the FM layer. Since, an increase
in the cooling field results in the growth of the FM cluster size and degree
of alignment, which advocates the increase in the values of tFM and MFM,
respectively. Also, higher cooling field enhances the values of interfacial
magnetization (SFM and SAFM) at the FM/AFM interface in the CSG phase [14].
Thus, as per the above equation 14, the HEB increases with the cooling field.
Although, at higher values the size of FM clusters further grows and despite
an increase in the values of tFM and MFM with the cooling field, a decrease in
the FM/AFM interface area reduces SFM and SAFM. Therefore, owing to the
increased value of the denominator and a reduction in the value of the
numerator of equation 14, we find a reduction in the values of HEB at higher
values of cooling fields, which is in good agreement with [14].

In order to further investigate the origin of the unconventional exchange bias
observed in the sample, we follow the relation given by Niebieskikwiat and
Salamon [70] at low temperatures ($\mu_{0}$Hcool $<$ kBT) below Tf, which
advocates that owing to the glassy behavior of surface spins, interface
moments ($m_{i}$ $\propto$ HEB) remain frozen and quantitative dependence of
HEB on the cooling field can be modelled such that [70, 14],

$\displaystyle\rm-H_{EB}\propto\frac{M_{E}}{M_{S}}\propto
J_{i}\left[\frac{J_{i}\mu_{o}}{(g\mu_{B})^{2}}L\left(\frac{\mu
H_{cool}}{k_{B}T_{f}}\right)+H_{cool}\right]$ (15)

where $J_{i}$, $g$, $L(x)$, $k_{B}$, $\mu_{B}$, and Hcool are the interface
exchange coupling constant, gyromagnetic factor, Langevin function, Boltzmann
constant, Bohr magneton, and cooling field. Here, $\mu$ is defined as the
magnetic moment of FM particles, i.e., $\mu=N_{FM}\mu_{0}$, where $N_{FM}$ is
the number of spins in the FM cluster. In the above equation, for the smaller
cooling field the first term dominates, i.e., -H$\rm{}_{EB}\propto J_{i}^{2}$
and for the larger values of the cooling field the second term dominates,
i.e., -H$\rm{}_{EB}\propto J_{i}$. Therefore, for a negative value of
$J_{i}<$0, the exchange bias parameters first increase with the cooling field
owing to the dominance of the first term ($J_{i}^{2}$) and then start
decreasing or a sign reversal is observed at the large values of the cooling
field, where the second term ($J_{i}$) in equation 15 dominates [71, 72]. We
fit the cooling field dependence of HEB and MEB using equation 15 [solid black
lines in Figs. 6(e, f)] where $J_{i}$ and $N_{FM}$ are considered as free
parameters. We obtain the values of $J_{i}=$ -10.24$\pm$0.22 meV and $N_{FM}=$
2.4$\pm$0.2 for the measuring field of $\pm$50 kOe and $J_{i}=$
-12.55$\pm$0.49 meV and $N_{FM}=$ 3.1$\pm$0.1 for the measuring field of
$\pm$90 kOe. Here, we find that the interface exchange coupling constant
($J_{i}$) higher as compared to reported in refs. [70, 14], which is
correlated to the observed giant values of exchange bias in the present
sample. Interestingly, the negative values of Ji explain the dominance of AFM
exchange interaction at the interface of FM/AFM interactions in the CSG
region, and smaller value of $N_{FM}$ manifest that the size of FM clusters is
small. Therefore, it advocates the larger interaction area between the AFM/FM
interface and further supports the observed giant values of exchange bias.
Moreover, a higher value of Ji and $N_{FM}$ for the measuring fields $\pm$90
kOe explains the decrease in the values of EB parameters at higher measuring
and cooling fields.

Figure 7: (a) The specific heat data (red circles) fitted in the temperature
range of 30–300 K (solid black line) using a combination of Debye and Einstein
models following the Padé approximation; blue solid line represents the value
according to the Dulong-Petit law (3nR), and (b) a best fit (solid blue line)
below 15 K including the electronic ($\gamma T$), lattice ($\beta T^{3}$), and
magnetic ($\delta T^{3/2}$) terms, inset highlights the deviation in the CP/T
versus T2 from a straight line in 2–8 K range.

Finally, we present the zero-field specific heat data [Cp(0, T)] in the range
of 1.9–300 K in Fig. 7(a), where the Cp value 111.7 J mole-1K-1) is found to
be less than the Dulong-Petit limit, CV = 3nR = 124.7 J mole-1K-1 (solid blue
line), where $n$ is the numbers of ions per mole (5 for this sample) and R is
the universal gas constant (8.314 J mole-1K-1), respectively [73]. We observe
that the specific heat value decreases with temperature and there is no
anomaly in the measured temperature range, which manifests an absence of any
long-range magnetic ordering in the sample. Also, there is no Shottky-like
anomaly in the present Nb substituted sample, in contrast to the reports in
La1-xSrxCoO3 [74, 75]. The observed specific heat is a cumulative effect of
contributions from the lattice due to phonons (Clattice), electronic because
of the conduction electrons (Celect), and a magnetic specific heat (Cmagn),
which can be written as Cp(T) = Clattice\+ Celect\+ Cmagn. First, we fit the
specific heat data only with the lattice contributions, i.e., from a
combination of Debye and Einstein heat-capacity models, using the formula
Clattice = mCV(Debye) \+ (1-m)CV(Einstein). The contribution of the Debye
heat-capacity model at a constant volume is given as [76],

$\displaystyle
C{\rm{}_{V(Debye)}}(T)=9nR\left(\frac{T}{\theta_{D}}\right)^{3}\int_{0}^{\theta_{D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx$
(16)

where $\theta_{D}$ is Debye temperature and its origin belongs to the acoustic
phonon vibrations in the sample and at a constant volume the contribution of
the Einstein heat-capacity model can be written as [76],

$\displaystyle
C{\rm{}_{V(Einstein)}}(T)=3nR\left(\frac{\theta_{E}}{T}\right)^{2}\frac{e^{\theta_{E}/T}}{(e^{\theta_{E}/T}-1)^{2}}$
(17)

where $\theta_{E}$ is the Einstein temperature corresponding to the optical
phonons. In Fig. 7(a), from the best fitting in 30-300 K range (solid black
line) using above equations we obtain a contribution of around 71% and 29%
from the Debye and Einstein models, respectively. Here, we use the Padé
approximation for the Debye model [77] for fitting the specific heat data. The
estimated values of $\theta_{D}$ and $\theta_{E}$ are = 688$\pm$10 K and
171$\pm$3 K, respectively. We find that the Nb substitution enhances the value
of $\theta_{D}$ as compared to La1-xSrxCoO3 (400-500 K) [74, 75]. This
enhancement in $\theta_{D}$ value can be associated with the non-saturating
value of the specific heat at room temperature [78]. The fitted curve is
extrapolated to the lowest measured temperature (dashed black line) in Fig.
7(a), which shows a deviation in the specific heat at low temperatures from
the lattice approximation, which manifests that an additional contribution is
required to understanding the low temperature heat capacity behavior.

Further, we observe that the specific heat exhibits a nonlinear behavior at
low temperatures, see the inset in Fig. 7(b). This indicates that the specific
heat does not follow $C_{p}=\gamma T+\beta T^{3}$ (where, $\gamma T$ and cubic
term $\beta T^{3}$ correspond to the electronic and lattice contributions,
respectively) at low temperatures and that is believed to be due to the short-
range magnetic interactions/glassy nature in the sample [35, 57]. Therefore,
we fit the Cp data including a magnetic term ($\delta T^{3/2}$) using
$C_{p}=\gamma T+\beta T^{3}+\delta T^{3/2}$ [57, 35]. The coefficient of the
electronic term ($\gamma$) can be utilized to estimate the density of states
(DOS) at the Fermi level, i.e., N(EF) as $\gamma=\pi^{2}k_{B}^{2}N(E_{F})$/3,
the coefficient of lattice contribution ($\beta$) is given by the formula
$\beta=(234nk_{B})/\theta_{D}^{3}$, or, $\theta_{D}^{3}=(12\pi^{4}nR)/5\beta$
[57, 79], and the coefficient of the magnetic term ($\delta$) can be used to
estimate the spin-wave stiffness constant ($D$), such that,
$\delta=0.113Ra^{3}(k_{B}/D)^{3/2}$, where $a$ is the lattice parameter of
elementary perovskite unit cell [76, 80]. The best fit (solid blue line) of
the low temperature $C_{p}$ using the above equation having three terms
(electronic+lattice+magnetic) in Fig. 7(b) provides a good description of the
data and the obtained fitting coefficients are $\gamma=$ 2.14$\pm$0.04 mJ
mole-1-K-2, $\beta=$ 0.24$\pm$0.01 mJ mole-1K-4, and $\delta=$ 0.13$\pm$0.02
mJ mole-1K-5/2. Here, the estimated value of DOS is found to be N(EF) =
5.47$\times$1023 eV-1mole-1 and the $\theta_{D}$ value is 323.7 K. This
$\theta_{D}$ value is nearly half of the value estimated from the Debye model
due to complex magnetic interactions at low temperatures, which can be
attributed to the cluster spin-glass behavior or the excitonic mass
enhancement in the low-lying crystal field [57]. Moreover, the estimated value
of spin-wave stiffness constant $D$ is 470 meVÅ2, which is in the range of
values reported for the perovskite manganites in [79, 80, 81].

##  IV  Conclusions

In conclusion, the effect of different external parameters (cooling field,
temperature, and measuring fields) on the exchange bias behavior in cluster
spin glass (CSG) La0.5Sr0.5Co0.85Nb0.15O3 is investigated in detail using the
field cooled isothermal M–H loops and magnetic training effect. The magnetic
memory effect and aging measurements confirm the presence of a glassy magnetic
state. Moreover, the detailed analysis of ac-$\chi$ data confirms the presence
of CSG with a characteristic spin-relaxation time-scale of $\tau_{0}$ =
8.4$\times$10-10 s. The observed CSG emerges due to the hole-poor and hole-
rich regions owing to the ferromagnetic (FM) double exchange interactions
(from Co4+-O-Co3+) and antiferromagnetic (AFM) super-exchange interactions
(from Co3+-O-Co3+ and Co4+-O-Co4+), respectively. Interestingly, we find that
the small size FM clusters are embedded in the AFM matrix like a conventional
FM/AFM interface and therefore results in a giant tunable exchange bias in the
sample. The EB parameters (HEB and MEB) show unusual dependency on the cooling
field. Further, we model this tunable behavior with the interface exchange
interaction model and find that the exchange interaction is strong and
negative $J_{i}=$ -10.24$\pm$0.22 meV and $N_{FM}=$ 2.4$\pm$0.2 (measuring
field of $\pm$50 kOe) and $J_{i}=$ -12.55$\pm$0.49 meV and $N_{FM}=$
3.1$\pm$0.1 (measuring field of $\pm$90 kOe). The specific heat data analysis
also provides insights into the complex magnetic interactions at low
temperatures.

##  V  Acknowledgements

RS acknowledges DST, INSPIRE for fellowship. RSD is grateful to the SERB–DST
for the financial support through core research grant (project reference no.
CRG/2020/003436).

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