# Unconventional superparamagnetic behavior in the modified cubic spinel
compound LiNi0.5Mn1.5O4

S. S. Islam  Vikram Singh  Somesh K  Prashanta K Mukharjee  School of Physics,
Indian Institute of Science Education and Research Thiruvananthapuram-695551,
India  A. Jain  S. M. Yusuf  Solid State Physics Division, Bhabha Atomic
Research Centre, Mumbai 400 085, India  R. Nath<EMAIL_ADDRESS>School
of Physics, Indian Institute of Science Education and Research
Thiruvananthapuram-695551, India

###### Abstract

Structural, electronic, and magnetic properties of modified cubic spinel
compound LiNi0.5Mn1.5O4 are studied via x-ray diffraction, resistivity, DC and
AC magnetization, heat capacity, neutron diffraction, 7Li nuclear magnetic
resonance, magnetocaloric effect, magnetic relaxation, and magnetic memory
effect experiments. We stabilized this compound in a cubic structure with
space group $P4_{3}32$. It exhibits semiconducting character with an
electronic band gap of $\Delta/k_{\rm B}\simeq 0.4$ eV. The interaction within
each Mn4+ and Ni2+ sub-lattice and between Mn4+ and Ni2+ sublattices is found
to be ferromagnetic (FM) and antiferromagnetic (AFM), respectively. This leads
to the onset of a ferrimagnetic transition at $T_{\rm C}\simeq 125$ K. The
reduced values of frustration parameter ($f$) and ordered moments reflect
magnetic frustration due to competing FM and AFM interactions. From the 7Li
NMR shift vs susceptibility plot, the average hyperfine coupling between 7Li
nuclei and Ni2+ and Mn4+ spins is calculated to be $\sim 672.4$ Oe/$\mu_{\rm
B}$. A detailed critical behaviour study is done in the vicinity of $T_{\rm
C}$ using modified-Arrott plot, Kouvel-Fisher plot, and universal scaling of
magnetization isotherms. The magnetic phase transition is found to be second
order in nature and the estimated critical exponents correspond to the 3D XY
universality class. A large magneto-caloric effect is observed with a maximum
value of isothermal change in entropy $\Delta S_{m}\simeq-11.3$ J/Kg-K and a
maximum relative cooling power of $RCP\simeq 604$ J/Kg for 9 T magnetic field
change. The imaginary part of the AC susceptibility depicts a strong frequency
dependent hump at $T=T_{\rm f2}$ well below the blocking temperature $T_{\rm
b}\simeq 120$ K. The Arrhenius behaviour of frequency dependent $T_{\rm f2}$
and the absence of ZFC memory confirm the existence of superparamagnetism in
the ferrimagnetically ordered state.

###### pacs:

##  I Introduction

Figure 1:  (a) Crystal structure of LNMO in three dimension. (b) Hyperkagome
lattice made of Mn4+ ions. (c) Pyrochlore lattice made of Ni2+ and Mn4+ ions.
(d) A section of the pyrochlore lattice showing corner sharing tetrahedras and
spin structure deduced from the neutron diffraction data at $T=5$ K.

Geometrically frustrated quantum magnets are long been a field of attraction
since they provide unique opportunity to realize novel quantum phases at low
temperatures.Ramiréz (1994); *Diep2013 One of the most studied geometrically
frustrated systems in three dimension (3D) is the antiferromagnetic (AFM)
pyrochlore lattice which features a network of corner sharing tetrahedras.
Prominent examples in this category are compounds with general formula
$AB_{2}$O4 (spinels), $A_{2}B_{2}$O7 (pyrochlores), and $AB_{2}$. In
particular, in spinel oxides $AB_{2}$O4, $B$ site ion forms a frustrated 3D
pyrochlore lattice. Owing to ground state degeneracy, these compounds have
witnessed various exotic low temperature phenomena ranging from quantum spin-
liquid, spin-glass, field induced transitions, magnetization plateaus to heavy
fermionic behaviour.Gardner  _et al._ (2010); Lee  _et al._ (2010); Lacroix
_et al._ (1996) In addition, there exists another series of compounds
$AB_{2}$X6, often referred as cubic modified pyrochlore lattice which mostly
contains either mixed-valent or two kinds of transition metal ions. The
compounds (Rb,Cs)Cr2F6,Ueda  _et al._ (2014) (K,Rb)Os2O6,Brühwiler  _et al._
(2004) CsW2O6,Hirai  _et al._ (2013) and CsNiCrF6Fennell  _et al._ (2019)
belong to this category and exhibit various exotic ground states.

Among the large class of spinel oxides, cubic LiMn2O4 (space group:
$Fd\bar{3}m$, Mn${}^{3+}\colon$ Mn${}^{4+}=1\colon 1$) is known to be a
celebrated high-voltage cathode material for rechargeable Li-ion
battery.Tarascon  _et al._ (1991) It is reported to have charge ordering
accompanied by orbital ordering due to the Jahn-Teller distortion in Mn3+ ions
and undergoes an antiferromagnetic (AFM) long-range-ordering (LRO) at low
temperatures.Tomeno  _et al._ (2001); *Sugiyama1187 In certain reports, the
compound is found to show a spin-glass (SG) behaviour without any magnetic LRO
which is attributed to the Mn3+/Mn4+ super-lattice charge order as well as the
effect of frustration.Jang  _et al._ (1999) These contradictory behaviours are
believed to be originated from the Mn site disorder.Zhang  _et al._ (2018)
Recently, coexistence of LRO and SG state is found in LiMn2O4 nanorods.Zhang
_et al._ (2018) The Ni doped LiNi0.5Mn1.5O4 (abbreviated as LNMO) crystallizes
in two different phases depending on the synthesis conditions. The
stoichiometric LNMO has $P4_{3}32$ space group and exhibits a 1$\colon$3
cation order of Ni2+ and Mn4+ ions, while non-stoichiometric LiNi0.5Mn1.5O4-δ
has $Fd\bar{3}m$ space group.Kim  _et al._ (2004)

In the $Fd\bar{3}m$ structure, Ni and Mn atoms randomly occupy one
crystallographic site while in the $P4_{3}32$ structure, they occupy two
inequivalent sites independently. In the $P4_{3}32$ structure which can also
be referred as modified cubic spinel, the edge sharing of MnO6 and NiO6
octahedras and the corner shared LiO4 tetrahedra lead to a complex three-
dimensional (3D) structure [see Fig. 1(a)]. When only the interaction among
the Mn4+ ions is considered, LNMO forms a 3D network of corner sharing Mn4+
triangles which is found to be a frustrated hyper-kagome lattice [see Fig. 1
(b)]. Further, when the interaction between Mn4+ and Ni2+ ions are taken into
account, a network of corner shared tetrahedras is formed where each
tetrahedra consists of three Mn4+ and one Ni2+ ions. Hence, the hyper-kagome
lattice transforms into a 3D pyrochlore lattice [see Fig. 1(c)]. Based on the
preliminary magnetic measurements, LNMO is reported to show a magnetic
transition at $T_{\rm C}\simeq 125$ K.Blasse (1966); *MukaiA672; *Amdouni100

In this paper, we present a detailed study of the physical properties of
stoichiometric modified cubic spinel compound LNMO ($P4_{3}32$). A
ferrimagnetic order is detected at $T_{\rm C}\simeq 125$ K. We found that the
interaction within each Mn4+ and Ni2+ sub-lattice is ferromagnetic (FM),
whereas the interaction between these two sublattices is antiferromagnetic
(AFM), which results in a ferrimagnetic behaviour below $T_{\rm C}$. Multiple
magnetic transitions are observed below $T_{\rm C}$, likely due to magnetic
frustration. It exhibits magnetic relaxation and magnetic memory effect below
$T_{\rm C}$, typically expected for superparamagnetic systems. A large
magnetocaloric effect (MCE) is obtained across the magnetic transition. The
critical analysis of magnetization and MCE data establish LNMO as a 3D XY type
magnet. The paper is organized in the following manner. The experimental
details concerning sample preparation and various measurements are described
in Sec. II. Section III contains the experimental results which includes
powder x-ray diffraction, resistivity, DC magnetization, heat capacity,
neutron diffraction, 7Li NMR, magnetocaloric effect, AC susceptibility,
magnetic relaxation, and magnetic memory effect measurements, followed by
discussions. Our experimental findings are summarized in Sec. IV.

##  II Methods

Traditional sol-gel synthesis method was adopted to synthesize LNMO in
polycrystalline form. At first, stoichiometric amount of lithium nitrate
(LiNO3, 99.99%), manganese nitrate tetra-hydrate [Mn(NO3)2.4H2O, $\geq 97$%],
and nickel nitrate hexa-hydrate [Ni(NO3)2.6H2O, 99.999%] were taken and
dissolved into ethanol. The mixture was continuously stirred at 80 0C until
the whole solvent is evaporated from the mixture and dark black coloured paste
was found. The resulting paste was then transferred into a crucible and
preheated at 500 0C for 2 hrs and then at 8000C for 8 hrs. Subsequently, the
furnace was switched off and the sample was cooled naturally within the
furnace. The resultant sample was found to be formed in the space group
$Fd\bar{3}m$, confirmed from the powder x-ray diffraction. In the next step,
the resultant sample was ground thoroughly and pressed into pellets. The
pellets were heated at 700 0C for 2 days and then cooled very slowly to room
temperature at a rate of 0.10C/min. This post firing of the $Fd\bar{3}m$ phase
sample at 700 0C was done to ensure the formation of the cation ordered
$P4_{3}32$ phase. This method is well established and already experimented
previously.Cai  _et al._ (2013); *Lee3118

Phase purity of the sample was checked from the high quality powder x-ray
diffraction (XRD) data, collected using a PANalytical x-ray diffractometer (Cu
Kα radiation, $\lambda_{\rm av}\simeq\ 1.5418$ Å). The temperature dependent
power x-ray diffraction was performed over a wide temperature range (15 K$\leq
T\leq 300$ K). For going below room temperature, an Oxford Phenix low-
temperature attachment to the diffractometer was used. To solve the magnetic
structure, temperature dependent neutron powder diffraction (NPD) experiment
was performed using the neutron powder diffractometer ($\lambda\simeq\ 1.094$
Å) with three linear position-sensitive detectors at Dhruva reactor, Bhabha
Atomic Research Center, India. Rietveld refinement of the powder XRD data and
NPD data was performed using FullPROF software package.Rodríguez-Carvajal
(1993)

The DC magnetization ($M$) was measured using a vibrating sample magnetometer
(VSM) attachment to a commercial Physical Property Measurement System (PPMS,
Quantum Design) as a function of temperature (2 K $\leq T\leq$ 600 K) and
magnetic field (0 to 9 T). For the high temperature measurements ($T\geq 380$
K), a high-$T$ oven was attached to the VSM. Similarly, AC susceptibility was
measured as a function of temperature (2 K $\leq T\leq$ 200 K) and frequency
(50 Hz $\leq\nu\leq$ 10 kHz) in an AC field of 5 Oe using ACMS option of the
PPMS. For the temperature dependent heat capacity ($C_{\rm p}$) measurement,
the relaxation technique was adopted and the measurement was carried out on a
pressed pellet using heat capacity option of the PPMS. Electrical resistivity
($\rho$) as a function of temperature was measured on a rectangular pellet
using the four probe technique in PPMS.

The NMR measurements were performed by employing pulsed NMR technique on 7Li
(nuclear spin $I=3/2$ and gyromagnetic ratio $\gamma_{N}/2\pi=16.546$ MHz/T)
nuclei over a wide temperature range (4 K$\leq T\leq 290$ K). For this
purpose, we have used a liquid helium cryostat (Janis, USA) with a field sweep
superconducting magnet and a Tecmag (Redstone) spectrometer. The spectral
measurements at different temperatures were carried out either by Fourier
transform of the NMR echo signal at a fixed field of $H=1.5462$ T or by
sweeping the field at a corresponding fixed frequency of 25.58 MHz.
Traditional saturation recovery pulse sequence was used to measure the 7Li
spin-lattice relaxation time ($T_{1}$).

##  III Results and Discussion

###  III.1 X-ray Diffraction

Figure 2 presents the XRD pattern of LNMO at two end temperatures 300 K and 15
K. To evaluate the unit cell parameters and atomic positions, Rietveld
refinement was performed on the powder XRD data. The initial structural
parameters for this purpose were taken from Ref. [Cai  _et al._ , 2013;
*Lee3118]. All the peaks could be successfully indexed with cubic non-
centrosymmetric space group $P4_{3}$32\. The refined unit cell parameters,
volume of the unit cell, and the atomic co-ordinates at room temperature are
listed in Table 1 which are in close agreement with the previous report. As
already described in Sec. I, LNMO exhibits 1:3 cation order, resulting in a
superstructure cubic $P4_{3}$32 space group. The cation ordering in LNMO can
be visualized by the emergence of several low angle and low intensity Bragg
peaks, such as (110), (210), (322), (410) etc.Branford  _et al._ (2002) These
peaks are not allowed in the Bragg reflections in the normal face-centered
cubic spinel ($Fd\bar{3}m$) structure, because of the reflection conditions
$(h+k)=2n$, $(h+l)=2n$, and $(k+l)=2n$. Here, $n$ is the integer and $(h,k,l)$
are the Miller indices. These low intensity peaks are highlighted in the inset
of the upper panel of Fig. 2. It can be seen that all these small Bragg
reflections could be perfectly indexed by using $P4_{3}$32 space group. We
have also tried to do the refinement with $Fd\bar{3}m$ space group, which
could not index these small Bragg peaks, thus confirming the phase purity of
the sample with $P4_{3}$32 space group.

Figure 2:  Upper panel: X-ray powder diffraction pattern (open circles) of
LNMO at room temperature. The red solid line is the Rietveld refinement fit
with $P4_{3}$32 space group. The expected Bragg positions are indicated by
green vertical ticks. Blue solid line at the bottom denotes the difference
between observed and calculated intensities. Inset: Enlarged portion of the
XRD pattern to visualize small Bragg peaks. Lower panel: Rietveld refinement
of the XRD pattern of LNMO at 15 K.

It is reported that the material synthesis following sol-gel method leads to
the formation of nano-crystalline form of LNMO.Kunduraci and Amatucci (2006);
*Wang32 To estimate the crystallite size, we fitted the XRD peaks at room
temperature by a Gaussian function and evaluated the full width at half maxima
(FWHM) of the individual peak. Subsequently, by using the Scherrer equation,
$\tau=K\lambda/\beta\rm{cos\theta}$ (where, $\tau$ is the crystallite size,
$K$ is the dimensionless shape factor which has a typical value of 0.9,
$\lambda$ is the x-ray wavelength, and $\beta$ is the line broadening at
FWHM), the average crystallite size is calculated to be $\sim 60$ nm.Patterson
(1939) Further, the analysis of the Scanning-Electron-Microscopy (SEM) data
also reveals the nano-crystalline nature of LNMO sample with average particle
size 100-150 nm.

Figure 3:  Variation of lattice constant ($a$) and unit cell volume ($V_{\rm cell}$) with temperature. The solid line represents the fit of $V_{\rm cell}(T)$ by Eq. (1). Table 1: Crystal structure data for LNMO at room temperature (cubic, space group: $P4_{3}32$). The obtained lattice parameters from the refinement are $a=b=c=8.1643(1)$ Å and $V_{\rm cell}\simeq 544.2~{}\AA^{3}$. Our fit yields quality factors $R_{\rm p}\simeq 10.1$, $R_{\rm wp}\simeq 6.66$, $R_{\rm e}\simeq 4.88$, and goodness of fit $\chi^{2}=[\frac{R_{\rm wp}}{R_{\rm e}}]^{2}\simeq 1.87$. Listed are the Wyckoff positions and the refined atomic coordinates ($x$, $y$, and $z$) for each atom. Atom | Site | $x$ | $y$ | $z$
---|---|---|---|---
Li | $8c$ | $-0.0002(4)$ | $-0.0002(4)$ | $-0.0002(4)$
Ni | $4b$ | $0.625$ | $0.625$ | $0.625$
Mn | $12d$ | $0.125$ | $0.3768(3)$ | $0.8732(3)$
O1 | $8c$ | $0.3816(7)$ | $0.3816(7)$ | $0.3816(7)$
O2 | $24e$ | $0.1509(6)$ | $-0.1395(9)$ | $0.1238(7)$

As shown in lower panel of Fig. 2, no extra peaks could be detected down to 15
K. Figure 3 depicts the temperature variation of lattice constant ($a$) and
unit cell volume [$V_{\rm cell}(T)$] obtained from the refinement. Both the
quantities are found to decrease systematically during cooling and neither any
structural transition nor any lattice distortion is observed in the entire
measured temperature range (15 K$\leq T\leq 300$ K). Following the method
described in Ref. Islam  _et al._ (2018), $V_{\rm cell}(T)$ is fitted by

$V_{\rm cell}(T)=\gamma U(T)/K_{0}+V_{0},$ (1)

where $V_{0}$ is the unit cell volume of the crystal structure at $T=0$ K,
$K_{0}$ is the bulk modulus, and $\gamma$ is the Gr$\ddot{\rm u}$neisen
parameter. $U(T)$ is the internal energy and it can be expressed in terms of
the Debye approximation as

$U(T)=9p\,k_{\rm B}T\left(\frac{T}{\theta_{\rm
D}}\right)^{3}\int_{0}^{\theta_{\rm D}/T}\dfrac{x^{3}}{e^{x}-1}dx.$ (2)

Here, $p$ is the number of atoms in the unit cell and $k_{\rm B}$ is the
Boltzmann constant. The parameters evaluated from the fitting are Debye
temperature $\theta_{\rm D}\simeq 965$ K, $\gamma/K_{0}\simeq 1.24\times
10^{-4}$ Pa-1, and $V_{0}\simeq 542.48$ Å3.

###  III.2 Resistivity

Temperature dependent electrical resistivity [$\rho(T)$] measured in zero
field is shown in Fig. 4. It increases rapidly with decreasing temperature
which indicates that the ground state is insulating in nature. Below 218 K,
$\rho(T)$ could not be measured since it exceeded the measurable range of the
instrument. To evaluate the activation energy, the temperature dependent
conductivity ($\sigma=1/\rho$) data were fitted by Arrhenius equation

$\sigma(T)=Aexp\left(-\dfrac{\Delta}{k_{\rm B}T}\right),$ (3)

where, $A$ is the proportionality constant and $\Delta$ is the activation
energy. In the inset of Fig. 4, ln($\sigma$) is plotted against $1/T$ to
highlight the activated behaviour. Our fit in the whole measured temperature
range (218 K $\leq$ $T$ $\leq$ 300 K) yields $\Delta/k_{\rm B}\approx 0.4$ eV.
This value of $\Delta/k_{\rm B}$ categorizes LNMO as a semiconductor.

Figure 4:  The electrical resistivity $\rho(T)$ of LNMO in zero field. Inset:
ln($\sigma$) vs $1/T$. The solid line is the fit using Eq. (3).

###  III.3 DC Magnetization

Figure 5:  Upper panel: Temperature dependent DC magnetic susceptibility
$\chi(T)$ at two different applied fields. Inset: $\chi(T)$ measured at 50 Oe
in ZFC and FC conditions. Lower panel: $1/\chi$ vs $T$ in $\mu_{0}H=0.5$ T.
Inset: Magnetization isotherm at $T=2$ K.

The upper panel of Fig. 5 presents the temperature dependent DC magnetic
susceptibility $\chi(T)$ ($\equiv M/H$) measured in an applied field of
$H=0.5$ T and 1.5 T. In the high temperature regime, $\chi(T)$ shows a gradual
increase with decreasing temperature. Below about 140 K, $\chi(T)$ increases
rapidly, indicating the onset of a ferrimagnetic/ferromagnetic ordering. From
the $d\chi/dT$ vs $T$ plot, the ordering temperature is found to be $T_{\rm
C}\simeq 125$ K. As depicted in the inset of the upper panel of Fig. 5, the
zero-field cooled (ZFC) and field cooled (FC) susceptibility data at $H=50$ Oe
show a significant bifurcation below $T_{\rm C}$. Such an irreversibility is a
characteristic behaviour of ferrimagnetic/ferromagnetic compoundsNath  _et
al._ (2010) and is also observed for various SGBag  _et al._ (2018, 2020) and
superparamagneticTsoi  _et al._ (2005) systems. Moreover, the ZFC $\chi(T)$
shows a well defined maxima at $T_{\rm b}\simeq 120$ K, which corresponds to
the blocking temperature, typically expected for a superparamagnet. A
significant difference in the behaviour of FC $\chi(T)$ is expected between a
superparamagnet and a SG system. For instance, FC magnetization always
increases for a superparamagnet whereas for a SG system, it either remains
flat or decreases with decreasing temperature, below $T_{\rm b}$.Sasaki  _et
al._ (2005); Bandyopadhyay and Dattagupta (2006) As noticed from Fig. 5, the
FC $\chi(T)$ below $T_{\rm b}$ increases monotonously with decreasing $T$,
which is a primary indication of the superparamagnetic blocking process.Sasaki
_et al._ (2005); Chen  _et al._ (2005) In order to confirm this behaviour, we
have performed a detailed ac susceptibility and magnetic memory effect
experiments which are discussed later.

The lower panel of Fig. 5 shows the inverse magnetic susceptibility ($1/\chi$)
for $\mu_{0}H=0.5$ T. In the paramagnetic regime ($T>T_{\rm C}$), $1/\chi$
typically shows a linear behaviour with temperature, due to uncorrelated
moments. In contrast, the observed high temperature non-linear behaviour with
a strong positive curvature is a possible signature of ferrimagnetic nature of
LNMO.Nath  _et al._ (2010); Kittel  _et al._ (1976) To extract the magnetic
parameters, we fitted the $\chi(T)$ data by the modified Curie-Weiss law

$\chi(T)=\chi_{0}+\frac{C}{T-\theta_{\rm CW}}.$ (4)

Here, $\chi_{0}$ is the temperature-independent susceptibility, $C$ is the
Curie constant, and $\theta_{\rm CW}$ is Curie-Weiss temperature. Our fit in
the high-temperature regime ($T\geq 450$ K) (see the lower panel of Fig. 5)
yields the parameters: $\chi_{0}\simeq 0.0015$ cm3/mol, $C\simeq 1.604$
cm3K/mol, and $\theta_{\rm CW}\simeq 144.4$ K. From the value of $C$, the
effective moment is calculated to be $\mu_{\rm eff}=\sqrt{3k_{\rm B}C/N_{\rm
A}\mu_{\rm B}^{2}}\simeq 3.58\mu_{\rm B}$, where $N_{\rm A}$ is the Avogadro’s
number. This is close to the expected spin-only value of $\mu_{\rm
eff}=\sqrt{\left(\dfrac{0.5\times[\mu_{\rm eff}(\rm
Ni^{2+})]^{2}+1.5\times[\mu_{\rm eff}(\rm
Mn^{4+})]^{2}}{0.5+1.5}\right)}~{}\mu_{\rm B}=3.64~{}\mu_{\rm B}$, taking
$\mu_{\rm eff}=2.83$ $\mu_{\rm B}$ and 3.87 $\mu_{\rm B}$ for Ni2+($S=1$) and
Mn4+($S=3/2$), respectively.Mukai and Sugiyama (2010) The positive value of
$\theta_{\rm CW}$ implies that the dominant interaction among the magnetic
ions is ferromagnetic (FM) in nature. The inset of the lower panel of Fig. 5
shows a complete magnetization isotherm ($M$ vs $H$) at $T=2$ K. It shows a
very weak hysteresis and the magnetization saturates quickly at $\mu_{0}H=0.5$
T, typically expected for a ferrimagnet. The saturation magnetization ($M_{\rm
s}$) is found to be $\sim 3.2~{}\mu_{\rm B}$. Assuming a two sub-lattice model
of magnetic species Ni2+ and Mn4+ with antiferromagnetic (AFM) coupling
between them and using molecular-field approximation, the saturation
magnetization for LNMO can be written as $M_{\rm s}=[S(\rm Mn^{4+})\times
g\times 1.5-S(\rm Ni^{2+})\times g\times 0.5]~{}\mu_{\rm B}$.Nath  _et al._
(2010); Kittel  _et al._ (1976) Taking $S(\rm Mn^{4+})=3/2$, $S(\rm
Ni^{2+})=1$, and $g=2$, $M_{\rm s}$ is calculated to be $3.5~{}\mu_{\rm B}$.
Usually, the value of $g$ for Mn4+ and Ni2+ is always more than 2 which should
produce $M_{\rm s}$ larger than $3.5~{}\mu_{\rm B}$.Sun  _et al._ (2016);
*Werner214414; *Ruan55 Clearly, our experimental value of $M_{\rm s}$ is
smaller than the expected spin-only value.

The extent of frustration in a spin system can be quantified by the
frustration ratio $f=\frac{|\theta_{\rm CW}|}{T_{\rm C}}$.Ramiréz (1994)
According to the mean field theory $\theta_{\rm CW}$ is nothing but the sum of
all exchange couplings present in the system i.e. $\theta_{\rm
CW}=\sum_{i}J_{i}$, where $i$ is the number of nearest neighbor spins.Domb and
Miedema (1964) Typically, for a non-frustrated AFM system $\theta_{\rm CW}\sim
T_{\rm C}$ (or $T_{\rm N}$) and $f$ has a value close to 1. However, for a
highly frustrated antiferromagnet, $f$ value is much larger than 1
($>10$).Ramiréz  _et al._ (2001) On the other hand, for a system having FM and
AFM interactions, the value of $\theta_{\rm CW}$ is reduced due to opposite
sign of the exchange couplings. This results in a decreased value of $f$. For
LNMO, the frustration ratio is calculated to be $f\simeq 144.4/125\simeq
1.16$. Since LNMO is having highly frustrated pyrochlore geometry, a reduced
value of $f$ clearly implies co-existence of AFM and FM interactions in the
system. Indeed, our neutron powder diffraction experiments (discussed later)
confirm this proposition.

###  III.4 Heat Capacity

Figure 6:  Upper panel: $C_{\rm p}(T)$ measured in zero magnetic field. The
dashed and solid lines represent the phonon ($C_{\rm ph}$) and magnetic
($C_{\rm mag}$) contributions, respectively. Inset: Enlarged view of low
temperature portion of $C_{\rm p}(T)$ data. The solid line is the fit as
described in text. Lower panel: $C_{\rm mag}/T$ and magnetic entropy ($S_{\rm
mag}$) vs $T$ along the left and right $y$-axes, respectively.

The temperature dependent heat capacity [$C_{\rm p}(T)$] measured in zero
field is shown in the upper panel of Fig. 6. A sharp and distinct peak is
observed at $T_{\rm C}\simeq 124$ K, indicating the magnetic transition. In
order to analyze the low temperature $C_{\rm p}(T)$ data, we first used the
relation $C_{\rm p}(T)=\beta T^{3}$, which did not fit the data effectively.
However, the low temperature $C_{\rm p}(T)$ data could be fitted nicely by
adding an extra term $\delta T^{3/2}$ to the above relation, i.e., $C_{\rm
p}(T)=\beta T^{3}+\delta T^{3/2}$. Here, the first term ($\beta T^{3}$)
represents the lattice contribution and the second term ($\delta T^{3/2}$) is
typical for ferromagnetic/ferrimagnetic and glassy systems.Gopal (2012);
Thomson and Thompson (1981) The inset of the upper panel of Fig. 6 depicts the
enlarged view of the low temperature portion of $C_{\rm p}(T)$. The solid line
is the fit using the above relation, in the temperature range 2-16 K. The
resulting $\beta$ and $\delta$ values are $\sim 1.07\times 10^{-4}$ J mol-1
K-4 and $\sim 0.0081$ J mol-1 K-5/2, respectively. The electronic contribution
$\gamma T$ is not considered in the fitting procedure, since LNMO is an
insulator at low temperatures.

For an estimation of the phonon contribution $C_{\rm ph}(T)$, we fitted the
experimental $C_{\rm p}(T)$ data in the high temperature regime ($T\geq
160~{}$K) by the Debye function

$C_{\rm ph}(T)=9R\left(\frac{T}{\theta_{\rm
D}}\right)^{3}\int_{0}^{\theta_{\rm D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx.$
(5)

Here, $R$ is the universal gas constant. The best fit was obtained with
$\theta_{\rm D}\simeq 735$ K which is close to the value obtained from the
$V_{\rm cell}(T)$ analysis. The high temperature $C_{\rm ph}(T)$ fit was
extrapolated down to low temperatures and subtracted from the experimental
$C_{\rm p}(T)$ data to obtain the magnetic contribution $C_{\rm mag}(T)$. The
obtained $C_{\rm mag}(T)$ is shown as a solid line in the upper panel of Fig.
6. $C_{\rm mag}(T)/T$ vs $T$ is presented in the left $y$-axis of the lower
panel of Fig. 6. The magnetic entropy $S_{\rm mag}(T)$ is calculated by
integrating the $C_{\rm mag}(T)/T$ in the whole measured temperature range as

$S_{\rm mag}(T)=\int_{2.1\rm K}^{T}\frac{C_{\rm mag}(T)}{T}dT.$ (6)

As shown in the lower panel of Fig. 6, the value of $S_{\rm mag}$ is found to
be $\sim 20.6$ J/mol-K at 250 K, which is very close to the theoretically
expected value of $S_{\rm mag}=0.5\times Rln[2S(\rm Ni^{2+})+1]$ \+ $1.5\times
Rln[2S(\rm Mn^{4+})+1]=21.8$ J/mol-K. One broad hump is observed in $C_{\rm
mag}(T)$ at around $T\sim 50$ K which cannot be attributed to any magnetic
transition as $\chi(T)$ does not show any feature at this temperature. Similar
feature is reported earlier for BaMn2As2 and BiMn2PO6 where it is proposed
that this broad hump is associated with the temperature dependent change in
the population of the Zeeman levels below transition, arising due to the
temperature dependent exchange field.Johnston  _et al._ (2011); Nath  _et al._
(2014) Using a Weiss molecular field theory on a Heisenberg model, it is
predicted that the hump like feature is pronounced for the systems with higher
spin values.Johnston  _et al._ (2011); Nath  _et al._ (2014) Thus, the
observed hump in $C_{\rm mag}(T)$ is obvious since LNMO has two high spins:
$S=1$ and $S=3/2$.

###  III.5 Neutron Diffraction

Figure 7:  Temperature evolution of the neutron powder diffraction pattern of
LNMO shown for the low angle regime. An enhancement in the intensity of
fundamental nuclear Bragg peaks is evident below $T_{\rm C}$. These Bragg
peaks at $2\theta\simeq 10.81\degree$, $13.31\degree$, $17.17\degree$, and
$18.84\degree$ correspond to (1,1,0), (1,1,1), (2,1,0), and (2,1,1) planes,
respectively.

Figure 7 depicts a series of neutron powder diffraction (NPD) patterns for
LNMO, over the temperature range $5-300$ K. An enhancement in the intensity of
fundamental nuclear Bragg peaks at $\sim 10.81\degree$, $13.31\degree$,
$17.17\degree$, and $18.84\degree$ are observed below $T_{\rm C}$. The
observation of additional intensity at the positions of fundamental nuclear
Bragg reflections (with no additional peaks) indicate the presence of a ferro-
or ferrimagnetic ordering. As shown in the upper panel of Fig. 8, all the
nuclear peaks at $T=140$ K could be refined using cubic crystal structure with
space group $P4_{3}32$. The refined structural parameters are listed in Table
2. These values are in close agreement with the refined values obtained from
the powder XRD data. The lower panel of Fig. 8 shows the Rietveld refinement
of NPD pattern at $T=5$ K. All the low angle peaks with additional intensity
could be indexed with a propagation vector $k=(0,0,0)$ and space group
$P4_{3}32$. The symmetry analysis shows that the observed NPD patterns can be
fitted assuming a collinear ferrimagnetic structure with magnetic moments
aligned along the [110] direction.

Figure 8:  Rietveld refined neutron powder diffraction patterns at $T=140$ K (upper panel) and $T=5$ K (lower panel). Open circles represent the experimental data, solid line represents the calculated curve, and difference between them is shown as a solid line at the bottom. Vertical marks correspond to the position of all allowed Bragg reflections for the nuclear (top row) and magnetic (bottom row) reflections. Inset: Temperature variation of ordered magnetic moment for Ni2+ and Mn4+ ions. Table 2: Structural parameters of LNMO refined from the neutron diffraction data at $T=140$ K (Structure: cubic and Space group: $P4_{3}32$). The obtained lattice parameters from the refinement are $a=b=c=8.2000(8)$ Åand $V_{\rm cell}\simeq 551.36(9)~{}\AA^{3}$. The fit yields quality factors $R_{\rm p}\simeq 21.5$, $R_{\rm wp}\simeq 18.2$, $R_{\rm e}\simeq 15.8$, and goodness of fit $\chi^{2}=[\frac{R_{\rm wp}}{R_{\rm e}}]^{2}\simeq 1.327$. Listed are the Wyckoff positions and the refined atomic coordinates ($x$, $y$, and $z$) for each atom. Atom | Site | $x$ | $y$ | $z$
---|---|---|---|---
Li | $8c$ | $0.0002(19)$ | $0.0002(19)$ | $0.0002(19)$
Ni | $4b$ | $0.625$ | $0.625$ | $0.625$
Mn | $12d$ | $0.125$ | $0.3787(11)$ | $0.8713(11)$
O1 | $8c$ | $0.3809(5)$ | $0.3809(5)$ | $0.3809(5)$
O2 | $24e$ | $0.1503(4)$ | $-0.1417(5)$ | $0.1298(5)$

The magnetic spin structure determined from Rietveld refinement of the NPD
pattern at $T=5$ K is shown in Fig. 1(d). Within each Ni2+ or Mn4+ sublattice,
the moments are arranged parallel to each other providing a FM intra-
sublattice interaction whereas between the Ni2+ and Mn4+ sublattices the
alignment is found to be antiparallel which provides an AFM inter-sublattice
interaction. From Fig. 1(d) it is clearly evident that the system is still
frustrated due to competing FM and AFM interactions. However, the extent of
frustration is definitely less than a conventional AFM pyrochlore lattice.
This indeed fall in line with the $\chi(T)$ analysis where the reduced value
of $f$ is attributed to the co-existence of AFM and FM interactions.

One can also understand the exchange interactions by looking at the bond
angles. From the refinement of NPD data at $T=5$ K, we found the angles
$\angle Mn-O1-Mn\simeq 93.8$ $\degree$, $\angle Mn-O2-Mn\simeq 98.4$
$\degree$, and $\angle Mn-O2-Ni\simeq 96.80$ $\degree$ and $95.3$ $\degree$.
According to Goodenough-Kanamori rule,Kanamori (1959); *Goodenough564 the
superexchange interaction between Mn4+ ($3d^{3}$) ions through the O2-
($2p^{2}$) ion is expected to be AFM when the angle $\angle Mn-O-Mn$ is linear
($\sim 180^{\degree}$) and it crosses over to FM interaction for the angle
$<135^{\degree}$.Shimakawa  _et al._ (1999) On the contrary, for the
interaction between Mn4+ ($3d^{3}$) and Ni2+ ($3d^{8}$) ions via O2-
($2p^{2}$) ion, it is reported that FM interaction occurs for angle $\angle
Mn-O-Ni$ close to $180^{\degree}$ and AFM interaction for the angle close to
$90^{\degree}$.Kanamori (1959) Thus, the obtained FM interaction between Mn4+
ions and AFM interaction between Mn4+ and Ni2+ ions are consistent with the
bond angle analysis using NPD data.

At $T=5$ K, the refined values of the ordered moment at $4b$ (Ni2+) and $12d$
(Mn4+) sites are found to be $\mu=1.60(1)$ $\mu_{\rm B}$ and 2.80(1) $\mu_{\rm
B}$, respectively, which are smaller as compared to the expected spin only
values (2 $\mu_{\rm B}/\rm{Ni^{2+}}$ for $S=1$ and 3 $\mu_{\rm
B}/\rm{Mn^{4+}}$ for $S=3/2$), assuming $g=2$. Such a reduced moment is
typically observed in low-dimensional and frustrated spin systems which is
attributed to the effect of quantum fluctuations and magnetic frustration,
respectively.Manna  _et al._ (2015) As shown in the inset of Fig. 8 the value
of the ordered moment for both Ni2+ and Mn4+ decreases as the temperature
approaches $T_{\rm C}$, typical behaviour of sub-lattice magnetization in the
ordered state. Using these values of the ordered moments at $T=5$ K, one
expects a saturation magnetization of $M_{\rm s}\simeq(\mu_{\rm Mn^{4+}}\times
1.5-\mu_{\rm Ni^{2+}}\times 0.5)~{}\mu_{\rm B}\simeq 3.4$ $\mu_{\rm B}$,
assuming a ferrimagnetic spin structure which is indeed consistent with
$M_{\rm s}\simeq 3.2$ $\mu_{\rm B}$ obtained from the $M$ vs $H$ curve at
$T=2$ K.

###  III.6 7Li NMR

Since Li is coupled strongly with the magnetic Mn4+ and Ni2+ ions, (see the
inset of Fig. 9) it is possible to get the information about the static and
dynamic properties of the spins by performing 7Li NMR.

####  III.6.1 7Li NMR Spectra

Figure 9:  Field-sweep 7Li NMR spectra at different temperatures for the
polycrystalline LNMO sample measured at 25.58 MHz. The vertical dashed line
corresponds to the 7Li resonance frequency of the non-magnetic reference.
Inset: Coupling of Li nucleus with four cubic units made of Ni2+ and Mn4+
ions. Figure 10:  Upper panel: Fullwidth at half maximum (FWHM) of 7Li NMR
spectra plotted as a function temperature. Inset: FWHM vs $\chi$ measured at
1.5 T is plotted with temperature as an implicit parameter. The solid line is
a linear fit. Lower panel: Temperature-dependent 7Li NMR shift ($K$) vs
temperature. Inset: 7Li NMR shift vs $\chi$ measured at 1.5 T is plotted with
temperature as an implicit parameter. The solid line is a linear fit.

For a quadrupolar 7Li ($I=3/2$) nucleus, one would expect two satellite peaks
along with the central line due to three allowed transitions. Our 7Li NMR
spectra, however, display a single spectral line in the whole temperature
range as shown in Fig. 9 which corresponds to the central transition
($+1/2\leftrightarrow-1/2$). The absence of satellite peaks could be either
due to low quadrupolar frequency or the distribution of the satellite peak
intensity over a wide frequency/field range. A single spectral line in 7Li NMR
is typically observed in low-dimensional oxides.Ranjith  _et al._ (2016);
*Ranjith024422 The line shape is found to be symmetric in the entire measured
temperature range suggesting the absence of magnetic anisotropy in the
compound. The NMR line broadens drastically below $T_{\rm C}$ which reflects
that Li nucleus senses the static internal field in the ordered state. With
decreasing temperature, the peak position of the spectral line is found to be
shifted. The upper panel of Fig. 10 depicts the fullwidth at half maximum
(FWHM) of 7Li NMR spectra plotted as a function temperature. At high
temperatures ($T\geq 200$ K), it is almost temperature independent, increases
abruptly below about 150 K (near $T_{\rm C}$), and then levels off to a
constant value with lowering temperature. The over all temperature dependent
behaviour of FWHM is similar to that of the bulk $\chi(T)$ data. The FWHM is
plotted against $\chi$ with temperature as an implicit parameter in the inset
of the upper panel of Fig. 10. It indeed gives a straight line behaviour over
the whole measured temperature range implying that the linewidth traces the
bulk $\chi(T)$.

The 7Li NMR shift ($K$) was extracted from the central peak positions of the
spectra in Fig. 9 by using the relation $K(T)=[H_{\rm ref}-H(T)]/H(T)$, where
$H(T)$ and $H_{\rm ref}$ are the resonance field of the sample and the
nonmagnetic reference sample, respectively. The temperature dependent NMR
shift [$K(T)$] is presented in the lower panel of Fig. 10. It shows a constant
behaviour at high temperatures and then increases abruptly below 150 K
suggesting the occurrence of a ferrimagnetic ordering. The overall behaviour
of $K(T)$ is again similar to the that observed for $\chi(T)$ and FWHM($T$).
Since $K(T)$ measures the intrinsic spin susceptibility $\chi_{\rm spin}(T)$,
one can write the relation

$K(T)=K_{\rm chem}+\frac{A_{\rm hf}}{N_{\rm A}\mu_{\rm B}}\chi_{\rm spin}(T),$
(7)

where $K_{\rm chem}$ is the temperature-independent chemical shift and $A_{\rm
hf}$ is the hyperfine coupling constant between the Li nuclei and electronic
(Ni2+, Mn4+) spins. The $K$ vs $\chi$ plot with temperature as an implicit
parameter is presented in the inset of the lower panel of Fig. 10. It produces
a linear behaviour in the entire measured temperature range. From the slope of
a straight line fit, the transfer hyperfine coupling is evaluated to be
$A_{\rm hf}\simeq 672.4$ Oe/$\mu_{\rm B}$.

####  III.6.2 Spin-lattice relaxation rate 1/$T_{1}$

Figure 11:  Spin-lattice relaxation rate $1/T_{1}$ vs temperature measured at
25.58 MHz. Inset: The longitudinal magnetization recovery curves at various
temperatures. The solid lines are the fits, as described in the text.

In order to probe the low energy spin dynamics, 7Li NMR spin-lattice
relaxation rate ($1/T_{1}$) is measured at the field corresponding to the
central peak position at different temperatures. The measurements are carried
out in the temperature range 4 K to 290 K. Recovery of the longitudinal
magnetization is monitored after a saturation pulse sequence. Some of the
representative recovery curves in the low temperature regime are shown in the
inset of Fig. 11. These recovery curves are fitted by single exponential
functionRanjith  _et al._ (2016)

$1-\frac{M(t)}{M_{0}}=Ae^{-t/T_{1}},$ (8)

where $M(t)$ is the nuclear magnetization at a time $t$ after the saturation
pulse and $M_{0}$ is the equilibrium magnetization. The $1/T_{1}$ extracted
from the fit is plotted with respect to the temperature in Fig. 11.

At high temperatures ($T\geq 160$ K), $1/T_{1}$ is almost temperature
independent. In the paramagnetic regime, where the temperature is higher than
the exchange energy between the spins, the temperature independent behaviour
of $1/T_{1}$ is obvious due to uncorrelated moments.Moriya (1956) Typically,
when one approaches the magnetic transition from high temperatures, $1/T_{1}$
is anticipated to show a sharp peak or divergence with temperature due to the
critical slowing down of the fluctuating moments. Our $1/T_{1}$ does show a
weak anomaly at $T_{\rm C}$ and then decreases rapidly below $T_{\rm C}$. The
decrease below $T_{\rm C}$ indicates the relaxation due to scattering of
magnons by the nuclear spins.Ranjith  _et al._ (2016)

###  III.7 Critical Scaling of Magnetization

Figure 12: (a) The Arrott plots ($M^{2}$ vs $H/M$) for LNMO at different
temperatures, above and below $T_{\rm C}$. (b) The modified Arrott plots
($M^{1/\beta}$ vs $(H/M)^{1/\gamma}$) for LNMO at different temperatures. The
solid lines are the linear fits to the data in the high field regime
($H\geq~{}2.5$ T) and are extrapolated to $H/M=0$. (c) Spontaneous
magnetization $M_{\rm S}$ and zero field inverse susceptibility
$\chi_{0}^{-1}$ as a function of temperature in the left and right $y$-axes,
respectively, obtained from the intercepts of the modified Arrott plots in the
vicinity of $T_{\rm C}$. The solid lines are the fits, as described in the
text. (d) The Kouvel-Fisher plots for $M_{\rm S}$ and $\chi_{0}^{-1}$. The
solid lines are the linear fits.

For a better understanding of the magnetic properties, we have performed the
critical behaviour study of magnetization in the vicinity of critical
temperature $T_{\rm C}$. Scaling hypothesis suggests that the second order
phase transition near $T_{\rm C}$ can be characterized by a set of critical
exponents ($\beta$, $\gamma$, and $\delta$) and magnetic equations of
state.Stanley (1971) Spontaneous magnetization $M_{\rm S}(T)$ at $T<T_{\rm
C}$, zero field inverse susceptibility $\chi_{0}^{-1}(T)$ at $T>T_{\rm C}$,
and isothermal magnetization ($M$ vs $H$) at $T=T_{\rm C}$ are related to the
critical exponents by the following equations:

$M_{S}(T)=M_{0}(-\varepsilon)^{\beta},\varepsilon<0,$ (9)
$\chi_{0}^{-1}(T)=\varGamma(\varepsilon)^{\gamma},\varepsilon>0,$ (10)
$M=XH^{1/\delta},\varepsilon=0,$ (11)

where $\varepsilon=(T-T_{\rm C})/T_{\rm C}$ is the reduced temperature and
$M_{0}$, $\varGamma$, and $X$ are the critical amplitudes.

Usually, the Arrott-Noakes equation of stateArrott and Noakes (1967) is
employed for a reliable estimation of critical exponents and $T_{\rm C}$ using
magnetic isotherms, which can be written as

$(H/M)^{1/\gamma}=A\varepsilon+BM^{1/\beta}.$ (12)

According to this equation, for a particular set of $\beta$ and $\gamma$
values, $M^{1/\beta}$ vs $(H/M)^{1/\gamma}$ plots should produce a set of
parallel straight lines in the high field region for different temperatures
around $T_{\rm C}$ and the isotherm at $T=T_{\rm C}$ should pass through
origin. The plot of $M^{1/\beta}$ vs $(H/M)^{1/\gamma}$ is often called the
modified Arrott plot (MAP) and the exponents reflect the universality class of
the spin system. The mean field exponents ($\beta=0.5$ and $\gamma=1$) lead to
conventional Arrott plot ($H/M$ vs $M^{2}$), which is traditionally used for
critical behaviour analysis.Arrott (1957) We have measured several magnetic
isotherms around the $T_{\rm C}$ in close temperature steps. To avoid residual
magnetization, at each temperature the measurements are done after cooling the
sample in zero field from high temperatures (above $T_{\rm C}$). Figure 12(a)
presents the standard Arrott plots for LNMO. All the curves show a non-linear
behaviour and a downward curvature in the high field region. This indicates
that the transition is non-mean field type and standard Arrott plot
(associated with mean field theory) may not be an appropriate way to analyze
the critical behaviour of this system. Further, the order of phase transition
can be determined from the slope of Arrott plots. According to the Banerjee
criterion, the positive slope corresponds to a second order phase transition
whereas the negative slope indicates a first order phase transition.Banerjee
(1964) Thus, the positive slope found in Fig. 12(a) confirms second order
nature of the transition.

Figure 13:  Magnetization isotherm ($M$ vs $H$) at $T\simeq T_{\rm C}$. Inset:
log$M$ vs log$H$ plot at $T\simeq T_{\rm C}$.

On the other hand, the MAPs constructed using the critical exponents of 3D
Heisenberg model resulted more linear behaviour in the high field region
compared the mean-field one. Therefore, we took the theoretical values of
$\beta$ and $\gamma$ for 3D Heisenberg model as the starting trial values to
construct MAPs. The linear fits to the MAPs in the high field regime were
extrapolated down to zero field and the values of $M_{S}(T)$ and
$\chi_{0}^{-1}(T)$ were obtained from the intercepts on the $M^{1/\beta}$ and
$(H/M)^{1/\gamma}$ axes, respectively. The temperature dependent $M_{\rm S}$
and $\chi_{0}^{-1}$ were fitted by Eqs. (9) and (10), respectively and the
values of $\beta$ and $\gamma$ were estimated. These set of $\beta$ and
$\gamma$ values were again used to construct a new set of MAPs. This whole
process was repeated several times, until we got a set of parallel straight
lines in the high field region with a set of stable $\beta$, $\gamma$, and
$T_{\rm C}$ values. The final MAPs are shown in Fig. 12(b). The obtained
$M_{\rm S}$ and $\chi_{0}^{-1}$ values are plotted as a function of
temperature in Fig. 12(c). The solid lines are the fits using Eqs. (9) and
(10), respectively. The fit using Eq. (9) yields $\beta=0.350(2)$ with $T_{\rm
C}=125.87(3)$ K and the fit using Eq. (10) yields $\gamma=1.320(2)$ with
$T_{\rm C}=125.83(3)$ K. These values are almost equal to the values obtained
from the final MAPs ($\beta=0.350$ and $\gamma=1.332$). As seen from Fig.
12(b), though straight lines are obtained in the high field region but
significant deviation from linearity was found in the low field region. This
is likely due to the averaging over magnetic domains which are mutually
misaligned.

Figure 14:  Upper panel: The reduced magnetization
$m={\mid\varepsilon\mid}^{-\beta}M(H,\varepsilon)$ vs reduced magnetic field
$h={H\mid\varepsilon\mid}^{-(\beta+\gamma)}$. The re-normalized curves in
different temperatures, just above and below $T_{\rm C}$ are collapsing into
two separate branches. Inset: The log-log plot of $m$ vs $h$ to magnify the
data in the low field region. Lower panel: $m^{2}$ vs $h/m$ plots where the
curves just above and below $T_{\rm C}$ are collapsing into two separate
branches.

In the next step, Kouvel-Fisher (KF) method is used to determine $\beta$,
$\gamma$, and $T_{\rm C}$ more accurately.Kouvel and Fisher (1964) The
equations involved in the KF method are

$M_{S}(T)\left[\dfrac{dM_{S}(T)}{dT}\right]^{-1}=(1/\beta)(T-T_{\rm C}),$ (13)

and

$\chi_{0}^{-1}(T)\left[\dfrac{d\chi_{0}^{-1}(T)}{dT}\right]^{-1}=(1/\gamma)(T-T_{\rm
C}).$ (14)

Table 3: Experimentally evaluated critical exponents ($\beta$, $\gamma$, and $\delta$) and $T_{\rm C}$ for LNMO obtained from MAP, KF plot, Critical Isotherm analysis, and MCE. For a comparison, the theoretically predicted values for different universality classes (mean-field, 3D Heisenberg, 3D Ising, and 3D XY) are also listed (taken from Ref. Kaul (1985)). Parameters |  MAP |  KV plot |  Critical Isotherm Analysis |  MCE |  Mean Field Model |  3D Heisenberg Model |  3D XY Model |  3D Ising Model
---|---|---|---|---|---|---|---|---
$\beta$ |  0.350(2) |  0.352(3) |  – |  – |  0.5 |  0.365 |  0.345 |  0.325
$\gamma$ |  1.320(2) |  1.314(2) |  – |  – |  1 |  1.386 |  1.316 |  1.241
$\delta$ |  4.771(2) |  4.733(4) |  4.77(3) |  4.72(9) |  3 |  4.80 |  4.80 |  4.82
$n$ |  0.61(2) |  0.61(3) |  – |  0.61(2) |  – |  – |  – |  –
$T_{\rm C}~{}(\rm K)$ |  125.83(3) |  125.84(8) |  125.6 |  – |  – |  – |  – |  –

From these two equations, it is apparent that $M_{\rm S}(T)[dM_{\rm
S}(T)/dT]^{-1}$ vs $T$ and $\chi_{0}^{-1}(T)[d\chi_{0}^{-1}(T)/dT]^{-1}$ vs
$T$ plots should yield straight lines with slopes $1/\beta$ and $1/\gamma$,
respectively. The beauty of this method is that, without any previous
knowledge about $T_{\rm C}$, one can estimate it from the intercept of the
straight line fits on the temperature axis. The KF plots for LNMO are
presented in Fig. 12(d). The critical exponent values evaluated from the KF
plots are $\beta=0.352(3)$ with $T_{\rm C}=125.83(6)$ K and $\gamma=1.314(2)$
with $T_{\rm C}=125.84(8)$ K. These values of critical exponents match nicely
with the ones obtained from the MAPs, indicating the reliability of the values
of critical exponents.

Figure 13 presents the critical isotherm at $T\simeq T_{\rm C}=125.6$ K.
According to Eq. $~{}\eqref{cre3}$, log$M$ vs log$H$ plot at the critical
temperature should produce a straight line with slope $1/\delta$. We have
plotted log$M$ vs log$H$ in the inset of Fig. 13 and a straight line fit
results $\delta=4.77(3)$. One can also estimate $\delta$ by using the Widom
relationWidom (1965)

$\delta=1+\dfrac{\gamma}{\beta}.$ (15)

By using the $\beta$ and $\gamma$ values from the KF method and MAPs, the
corresponding $\delta$ value is estimated to be $\delta=4.733(4)$ and
$\delta=4.771(2)$, respectively. It is found that the $\delta$ value obtained
from the Widom relation and the critical isotherm analysis are very close to
each other, reflecting the self-consistency of our $\beta$ and $\gamma$
estimations.

According to scaling hypothesis,Kaul (1985) the obtained critical exponents
should follow the universal scaling function

$M(H,\varepsilon)=\varepsilon^{\beta}f_{\pm}\left(\dfrac{H}{{\mid\varepsilon\mid}^{\beta+\gamma}}\right),$
(16)

where $f_{+}$ and $f_{-}$ are the regular functions for $T<T_{\rm C}$ and
$T>T_{\rm C}$, respectively. Equation (16) can further be simplified in terms
of reduced magnetization ($m$) and reduced field ($h$) as

$m=f_{\pm}(h),$ (17)

where $m={\mid\varepsilon\mid}^{-\beta}M(H,\varepsilon)$ and
$h=H{\mid\varepsilon\mid}^{-(\beta+\gamma)}$. Equations (16) and (17) suggest
that for appropriate choice of critical exponents, all the curves in the $m$
vs $h$ and $m^{2}$ vs $h/m$ plots should fall into two separate branches:
$f_{+}$ for $T>T_{\rm C}$ and $f_{-}$ for $T<T_{\rm C}$. This behaviour can be
clearly visualized in Fig. 14. In the inset of the upper panel of Fig. 14 we
have plotted $m$ vs $h$ in log-log scale which magnify the data in the low
field regime and confirms no dispersion among the curves in two branches. This
further confirms the reliability of the estimated critical exponent values.

For a comparison, all the values of critical exponents evaluated by the above
methods along with the theoretical values corresponding to various
universality classes (mean-field, 3D Heisenberg, 3D Ising, and 3D XY) are
listed in Table 3. One can see that our estimated critical exponents
($\beta\sim 0.35$, $\gamma\sim 1.32$, and $\delta\sim 4.772$) are very close
to the 3D XY-Model. This implies that LNMO belongs to the 3D-XY universality
class. This is quite consistent with the magnetic structure deduced from the
NPD data where the spin alignments are restricted only to the $ab$-plane.
There are certain compounds known to show the evidence of magnetic LRO
belonging to the 3D-XY universality class e.g. CuGeO3Lumsden  _et al._ (1996),
Gd2IFe2, Gd2ICo2, and Gd2BrFe2Reisser  _et al._ (1995).

###  III.8 Magnetocaloric Effect

Figure 15:  (a) The isothermal change in magnetic entropy ($\Delta S_{\rm m}$)
as function of temperature ($T$) and field ($H$). (b) The adiabatic change in
temperature ($\Delta T_{\rm ad}$) as a function of temperature ($T$) and field
($H$). (c) Universal curve plot of normalized entropy change as a function of
$\theta$ with the application of 1 T to 9 T magnetic field change. (d)
Absolute value of entropy change at the peak position $\Delta S_{\rm m}^{\rm
peak}$ and relative cooling power ($RCP=\Delta S_{\rm m}^{\rm
peak}\times\delta T_{\rm FWHM}$) as a function of magnetic field in the left
and right $y$-axes, respectively. Solid lines are the fits as described in the
text.

The MCE is defined as the reversible change in temperature of a magnetic
material while magnetic field is applied or removed in an adiabatic condition.
Generally, isothermal entropy change ($\Delta S_{\rm m}$) and adiabatic
temperature change ($\Delta T_{\rm ad}$) with respect to the change in
magnetic field quantify the MCE of a system. From the Maxwell’s thermodynamic
relation, $(\partial S/\partial H)_{T}=(\partial M/\partial T)_{H}$, $\Delta
S_{\rm m}$ can be estimated using the magnetization isotherm ($M$ vs $H$) data
as

$\Delta S_{\rm m}(H,T)=\int_{H_{\rm i}}^{H_{\rm f}}\dfrac{dM}{dT}dH.$ (18)

Figure 15(a) presents the 3D plot of $\Delta S_{\rm m}$ with the change in
field ($H$) and temperature ($T$). $\Delta S_{\rm m}$ changes gradually with
$H$ and shows a maximum entropy change across the $T_{\rm C}$, with a highest
value of $\Delta S_{\rm m}\simeq 11.2$ J/kg-K for 9 T field change.
Furthermore, $\Delta T_{\rm ad}$ is calculated from the zero field heat
capacity and magnetization isotherms using the relation

$\Delta T_{\rm ad}=-\int_{H_{\rm i}}^{H_{\rm f}}\dfrac{T}{C_{\rm
p}}\dfrac{dM}{dT}dH.$ (19)

The 3D plot of $\Delta T_{\rm ad}$ as a function of $H$ and $T$ is shown in
Fig. 15(b). $\Delta T_{\rm ad}$ shows a gradual temperature evolution with
respect to $H$ showing a maximum value $\sim 4$ K for 9 T field change near
$T_{\rm C}$. The shape of both $\Delta S_{\rm m}(T)$ and $\Delta T_{\rm
ad}(T)$ peaks appear to be asymmetric and expanded over a wide temperature
range around $T_{\rm C}$.

Another important parameter which is very useful to evaluate the performance
of a magneto-caloric material is relative cooling power ($RCP$). It defines
the amount of heat transfer between it’s hot and cold reservoirs which can be
written mathematically as

$RCP=\int_{T_{\rm cold}}^{T_{\rm hot}}\Delta S_{\rm m}(T,H)dT.$ (20)

Here, $T_{\rm cold}$ and $T_{\rm hot}$ are the temperatures corresponding to
the cold and hot reservoirs, respectively. Thus, $RCP$ can be evaluated
approximately as

$\arrowvert RCP\arrowvert_{\rm approx}=\Delta S_{\rm m}^{\rm peak}\times\delta
T_{\rm FWHM},$ (21)

where $\Delta S_{\rm m}^{\rm peak}$ is the peak value and $\delta T_{\rm
FWHM}$ is the full-width at half maximum (FWHM) of the $\Delta S_{\rm m}(T)$
curves in Fig. 15(a). The highest value $RCP$ is found to be $\sim 604$ J/Kg
for 9 T field change. This value of $RCP$ is quite high and comparable to
other well known magneto-caloric materials, having $T_{\rm C}$ around this
temperature.

In Table 4, we have made a comparison of the $RCP$ and $\Delta S_{\rm m}^{\rm
peak}$ values of LNMO with some other magneto-caloric materials with $T_{\rm
C}=110-140$ K. In most of the materials with second order phase transition,
the shape of the $\Delta S_{\rm m}$ curves is broad and asymmetric which is
one of the main reasons for the large value of $RCP$. Though, materials with
first order phase transition show giant MCE and large $\Delta S_{\rm m}$ and
$\Delta T_{\rm ad}$ values but the width of these peaks are narrow, which
restricts the usability of these materials for a cyclic operation. Another
drawback is that the materials with first order phase transition show
hysteretic behaviour which leads to energy loss.Franco  _et al._ (2012)
Therefore, materials with second order phase transition are more preferred for
the magnetic refrigeration purpose than the ones with first order phase
transition. It has been theoretically predicted that geometrically frustrated
magnets could show enhanced MCE compared to the ordinary non-frustrated
magnets.Zhitomirsky (2003) In particular, pyrochlore lattices are predicted to
show highest MCE among the geometrically frustrated lattices. Subsequently
this idea has been utilized to design several new magnetocaloric materials
with strong spin fluctuations and/or frustration.Tishin and Spichkin (2016);
Sosin  _et al._ (2005); *Das104420 Indeed, LNMO is a frustrated pyrochlore
magnet and hence the asymmetric behaviour can be attributed to the effect of
magnetic frustration. Thus, the obtained large $\Delta S_{\rm m}$ and $RCP$
values make LNMO a promising compound for magnetic refrigeration purpose.

Table 4: Comparison of experimental $\Delta S_{\rm m}^{\rm peak}$ and $RCP$ values for LNMO with some reported magneto-caloric materials having $T_{\rm C}=110-140$ K for a field change of 5 T. system |  $T_{\rm C}$ (K) | Nature of transition |  $\Delta S_{\rm m}^{\rm peak}$ (J/Kg-K) |  $RCP$ (J/Kg) | Ref.
---|---|---|---|---|---
Tb2NiMnO6 | 110 | Second order | 5.2 | - | Chakraborty  _et al._ (2017)
DyGa | 113 | Second order | 5.8 | 381.9 | Zheng  _et al._ (2012)
LiNi0.5Mn1.5O4 | 125.8 | Second order | 7.76 | 302 | This work
GdCo0.2Mn1.8 | 140 | Second order | 4.11 | 320 | Zhang  _et al._ (2007)
Tb5Ge2-xSi2-xMn2x, $2x=0.1$ | 123 | First order | 20.84 | 330.43 | Yuzuak  _et al._ (2010)
Dy(Co0.98 Ni0.02)2 | 126 | First order | 11 | 304 | Balli  _et al._ (2007)
YFe2H4.2 | 132 | First order | 7.11 | 263 | Paul-Boncour and Mazet (2009)

Further, MCE can also be utilized to gain more insight about the nature of the
magnetic phase transition and the critical exponents. The universal scaling
curve construction of $\Delta S_{\rm m}$ proposed by Franco et. al.Franco  _et
al._ (2006); *Franco093903 is generally used for this purpose. This method is
tested on variety of materials and found to be a very efficient way of
investigating the nature of the phase transition.Bonilla  _et al._ (2010) To
construct the universal scaling curves, first we normalized all the $\Delta
S_{\rm m}$($T$) curves with their respective peak values (i. e. $\Delta S_{\rm
m}$/$\Delta S_{\rm m}^{\rm peak}$) for each field change and then plotted as a
function of $\theta$ in Fig. 15(c). Here, $\theta$ is the re-scaled
temperature, which is given by

$\theta=-(T-T_{\rm C})/(T_{\rm r1}-T_{\rm C})~{}{\rm for}~{}T\leq T_{\rm C}$
(22)

and

$\theta=~{}(T-T_{\rm C})/(T_{\rm r2}-T_{\rm C})~{}~{}{\rm for}~{}T>T_{\rm C}.$
(23)

In the above, $T_{\rm r1}$ and $T_{\rm r2}$ are the two reference temperatures
corresponding to $0.5\times\Delta S_{\rm m}^{\rm peak}$ values and $T_{\rm
C}\simeq 125.8$ K [obtained from the Kouvel-Fisher plot]. As it is seen from
Fig. 15(c), all the normalized curves for different field changes collapse
into a single curve similar to the other reported compounds showing second
order phase transition.Bonilla  _et al._ (2010); Singh  _et al._ (2020)

The $\Delta S_{\rm m}^{\rm peak}$ and $RCP$ values are plotted in Fig. 15(d),
as a function of field in the left and right $y$-axes, respectively. For the
purpose of critical analysis, we have fitted the field dependent $\Delta
S_{\rm m}^{\rm peak}$ and $RCP$ curves by the following power lawsFranco  _et
al._ (2006); *Franco093903

$\lvert\Delta S_{\rm m}^{\rm peak}\arrowvert\propto H^{n}$ (24)

and

$\arrowvert RCP\arrowvert\propto H^{(1+1/\delta)}.$ (25)

The fit of Eq. (24) to the $\Delta S_{\rm m}^{\rm peak}$ vs $T$ data yields
$n=0.61(2)$. The exponent $n$ is related to the critical exponents $\beta$ and
$\gamma$ by the relation

$n=1+\left(\dfrac{\beta-1}{\beta+\gamma}\right).$ (26)

Using $\beta$ and $\gamma$ values obtained from the modified Arrot plot and
Kouvel-Fisher plots, the value of $n$ is estimated to be 0.61(2) and 0.61(3),
respectively. These values of $n$ obtained from various analysis methods are
consistent with each other and confirms the reliability of our critical
analysis technique. Similarly, the fit of $RCP$ vs $H$ data using Eq. (25)
yields $\delta=4.72(9)$, which is consistent with the values obtained from
modified Arrrott plot, Kovel-Fisher plot, and critical isotherm analysis
techniques.

Figure 16:  Field and temperature dependence of the exponent $n$ using Eq.
(27).

A more quantitative analysis of the phase transition can be done by using the
method proposed recently by Law et. al.Law  _et al._ (2018). The exponent $n$
in Eq. (24) is normally field and temperature dependent and can also be
calculated locally as

$n(T,H)=\dfrac{{\rm d~{}ln}\arrowvert\Delta S_{\rm m}\arrowvert}{{\rm
d~{}ln}H}.$ (27)

We used Eq. (27) to quantify the local $H$ and $T$ variation of exponent $n$
and plotted as a 3D plot in Fig. 16. Since in the low field range, the system
has multi-domain states, we have shown only the values corresponding to high
fields ($H>2$ T) in Fig. 16. At low temperatures ($T<T_{\rm C}$), the $n$ vs
$T$ curve is found to be almost constant and $n$ approaches a value $\sim
0.94$ which is close to 1. As the temperature is increased, $n$ decreases
smoothly and passes through a minima at $T\simeq T_{\rm C}$ with $n=0.61(1)$.
At high temperatures, it increases almost linearly and reaches upto a value
$n=1.98$ which is close to 2. At $T=T_{\rm C}$, the value of $n$ depends on
the value of other critical exponents of the material, as they are connected
by Eq. (26). The overall behaviour of $n$ vs $T$ curve (with exponent value,
$n<2$) is found to be similar to other compounds showing second order phase
transition.Law  _et al._ (2018)

###  III.9 AC Susceptibility

Figure 17:  Upper panel: Real part of the AC susceptibility $\chi^{\prime}$ vs
temperature at different frequencies. The arrow points to the change in
$\chi^{\prime}$ with frequency. Lower panel: Imaginary part of the AC
susceptibility $\chi^{\prime\prime}$ vs temperature at different frequencies.
Figure 18:  Enlarged view of $\chi^{{}^{\prime\prime}}(T)$ at low temperatures
and at different frequencies. The curves are vertically translated to get a
clear view of the peak positions. The vertical arrows guide the peak positions
with frequency.

Figure 17 presents the temperature dependent AC susceptibility measured in
different frequencies and at a fixed AC field of $H_{\rm AC}\simeq$ 5 Oe. As
shown in the upper panel of Fig. 17, the real part of the AC susceptibility
($\chi^{\prime}$) shows a sharp peak at $T_{\rm C}\simeq 125$ K and below 50 K
it decreases with temperature, akin to the behaviour observed in the ZFC DC
$\chi(T)$ (inset of Fig. 5). The peak at 125 K is almost frequency independent
indicating the onset of long-range ferrimagnetic order. Another broad feature
appears at around 25 K in low frequencies and the absolute value of
$\chi^{\prime}$ at this temperature is found to be reduced with increasing
frequency (indicated by a downward arrow). The imaginary part of the AC
susceptibility ($\chi^{{}^{\prime\prime}}$) is shown in lower panel of Fig. 17
as a function of temperature. Similar to the $\chi^{\prime}(T)$ data,
$\chi^{{}^{\prime\prime}}(T)$ also shows a sharp peak at $T_{\rm C}$ which is
frequency independent. At low temperatures, $\chi^{{}^{\prime\prime}}(T)$
shows various anomalies and some of them are frequency dependent.

To clearly visualize those low temperature features,
$\chi^{{}^{\prime\prime}}(T)$ data at different frequencies are vertically
translated and magnified in Fig. 18. At the lowest measured frequency of 0.2
kHz, $\chi^{{}^{\prime\prime}}(T)$ below $T_{\rm C}$ exhibits two broad humps
at $\sim 72$ K and $\sim 20$ K and a narrow peak at $\sim 3.5$ K, as indicated
by the upward arrows. The broad hump at $T_{\rm f1}\sim 72$ K disappears as
the frequency is increased beyond 0.5 kHz. With a very careful observation, we
found that the broad hump near 20 K consists of two shoulder like features,
one at $T_{\rm f2}\simeq 18$ K and another at $T_{\rm f3}\simeq 20$ K. As the
frequency is increased, the shoulder at $T_{\rm f2}$ becomes more pronounced
and shifts towards high temperatures, reflecting a glassy behaviour. On the
other hand, the shoulder at $T_{\rm f3}$ remains frequency independent and
gets diminished at high frequencies. Similarly, the peak at $T_{\rm f4}\sim
3.5$ K also remains unaltered with frequency indicating that it could be an
AFM transition. The multiple magnetic transitions in the ordered state can be
attributed to the effect of magnetic frustration.Dho  _et al._ (2002) The
absence of these features in $\chi^{\prime}$ can be believed as a consequence
of large value of $\chi^{\prime}$ associated with ferrimagnetism.Mahendiran
_et al._ (2003)

Since all the peaks are frequency independent except one, we focused our
analysis only on the frequency dependent peak at $T_{\rm f2}$. The peak
positions of $T_{\rm f2}$ and $T_{\rm f3}$ are extracted by a double Gaussian
fit (shown in Fig. 18 for 3.5 kHz). The relative shift of freezing temperature
($T_{\rm f2}$) with frequency ($\nu$) can be quantified by calculating the
Mydosh parameter $\delta T_{f}$, given byMydosh (2014)

$\delta T_{f}=\left(\dfrac{\Delta T_{\rm f}}{T_{\rm f}\Delta(\rm
log_{10}\nu)}\right),$ (28)

where $\Delta T_{\rm f}=(T_{\rm f})_{\nu_{1}}-(T_{\rm f})_{\nu_{2}}\simeq
16.65$ K and $\Delta\rm log_{10}(\nu)=\rm log_{10}(\nu_{1})-\rm
log_{10}(\nu_{2})$. Here, $\nu_{1}$ and $\nu_{2}$ are taken as the lowest (200
Hz) and highest (10 kHz) measured frequencies, respectively. For LNMO, the
Mydosh parameter is estimated to be $\delta T_{f}\simeq 0.4$, which is found
to be two orders of magnitude higher than the value for canonical SG systems
[e.g. AuMn, ($\delta T_{\rm f}\simeq 0.0045$)]Mulder  _et al._ (1982) and one
order magnitude higher than the cluster SG systems [e.g. Cr0.5Fe0.5Ga,
($\delta T_{\rm f}\simeq 0.017$)].Bag  _et al._ (2018) However, it lies in the
range of a superparamagnet [e.g. $\alpha-$(Ho2O3B2O3), ($\delta T_{\rm
f}\simeq$ 0.28)].Mydosh (2014)

Figure 19:  Upper panel: ln$(\tau)$ vs 1/$T_{\rm f2}$ plot. The solid line is
the fit using Eq. (29) in the high temperature region. Lower panel: $\rm
log_{10}(\tau)$ vs $\rm log_{10}(T_{\rm f2}/T_{\rm g}-1)$ plot. The solid line
is the fit using Eq. (31). Lower inset: $T_{\rm f2}$ vs $\rm log_{10}\nu$ plot
in the low temperature region. The solid line is the fit as described in the
text.

To understand the frequency dependence of $T_{\rm f2}$ and the correlation
among the magnetic entities we analyzed the data using various theoretical
models. At first, we tried to fit the data using Arrhenius lawBinder and Young
(1986)

$\tau=\tau_{0}~{}{\rm exp}\left(\dfrac{E_{\rm a}}{k_{\rm B}T_{\rm f}}\right),$
(29)

where $\tau$ is the time scale of dynamical fluctuation, $\tau_{0}$ is the
relaxation time for the single spin-flip, $E_{\rm a}/k_{\rm B}$ is the
activation energy required to overcome the energy barrier by which the meta-
stable states are separated, and $\tau_{0}$ is the time taken to overcome the
energy barrier. In the upper panel of Fig. 19, we have plotted ln$\tau$ vs
$1/T_{\rm f2}$. Clearly, it shows a change in slope at around 27 K. In the
high temperature ($T>27$ K) region, the data could be fitted well by Eq. (29)
resulting $E_{\rm a}/k_{\rm B}\simeq 61.7$ K and $\tau_{0}\simeq 3.7\times
10^{-6}$ s. Such an Arrhenius behaviour is often considered to be the
characteristic feature of superparamagnetism.Suzuki  _et al._ (2001) The value
of $\tau_{0}$ is also seems to be within range expected for superparamagnets
($10^{-6}$ to $10^{-9}$ s)Kumar  _et al._ (2017).

In the low temperature ($T<27$ K) region, the data show a significant
deviation from the Arrhenius law as evident in the upper panel of Fig. 19.
This indicates that there are two different relaxation mechanisms involved. We
fitted the low-$T$ data by the dynamical scaling law or power law, predicted
for SG systemsSouletie and Tholence (1985)

$\tau=\tau^{\prime}~{}{\rm exp}\left(\dfrac{T_{\rm f2}}{T_{\rm
g}}-1\right)^{-z\nu^{\prime}}.$ (30)

Here, $\tau^{\prime}$ has the same physical meaning as $\tau_{0}$, $T_{\rm g}$
is the freezing temperature as $\nu$ approaches zero, and $\tau^{\prime}$ =
$\xi^{\rm z}$ with $z$ being the dynamical critical exponent. The correlation
length has the form $\xi=(T_{\rm f2}/T_{\rm g}-1)^{-\nu^{\prime}}$ with
critical exponent $\nu^{\prime}$. Equation (30) can be rewritten in a
simplified form as

${\rm log_{10}}\tau={\rm log_{10}}\tau^{\prime}-z\nu^{\prime}{\rm
log_{10}}\left(\dfrac{T_{\rm f2}}{T_{\rm g}}-1\right).$ (31)

As shown in the lower panel of Fig. 19, Eq. (31) fits well to the low-$T$ data
giving $\tau^{\prime}\simeq 5.08\times 10^{-9}$ s and
$z\nu^{\prime}\simeq(6.6\pm 0.2)$. In the fitting process we have fixed
$T_{\rm g}\simeq(21.03\pm 0.01)$ K, obtained from the $y$-intercept of the
linear fit of $T_{\rm f2}$ vs log${}_{10}\nu$ plot (see the inset of lower
panel of Fig. 19). The larger value of $\tau^{\prime}$ clearly indicates that
the spin dynamics is slower than conventional SG systems ($\sim 10^{-13}$
s).Mydosh (2014) Such a high value of $\tau^{\prime}$ is previously reported
for various reentrant-SG systems.Kumar  _et al._ (2017); Viswanathan and Kumar
(2009); *Hanasaki086401 Further, the value of z$\nu^{\prime}$ also falls in
the range expected for typical SG systems ($z\nu^{\prime}\simeq 4\textendash
12$) and comparable to the values reported for various reentrant-SG
systems.Kumar  _et al._ (2017); Viswanathan and Kumar (2009); *Hanasaki086401

Thus, the frequency dependence of $T_{\rm f2}$ follows an Arrhenius behaviour
at higher temperatures which could suggest that the system is superparamagnet
far above the critical region ($T\gg T_{\rm g}$). As we move closer to $T_{\rm
g}$, Arrhenius behaviour breaks down and power law behaviour takes over which
evidences the existence of SG transition at $T_{\rm g}\simeq 21.03$ K, similar
to the dilute magnet LiHoxY1-xF4 ($x=0.045$).Quilliam  _et al._ (2008) In
order to resolve this ambiguity, memory effect measurements are discussed
later.

###  III.10 Nonequilibrium Dynamics

####  III.10.1 Magnetic Relaxation

Figure 20:  Time dependence of ZFC magnetization at different temperatures
($T=2,10,\rm{and}~{}70$ K) in $H=200$ Oe and after a waiting time of 60 s. The
solid lines are the fits using Eq. (32).

To explore the low temperature spin dynamics in LNMO, we have performed the
magnetic relaxation time measurement at different temperatures (2 K, 10 K, and
70 K), below $T_{\rm b}$. The sample was cooled in ZFC condition from high
temperature paramagnetic state ($T\geq 200$ K) to the desired temperature.
After waiting for $t_{\rm w}=60$ s at the desired temperature, a small
magnetic field of $H\simeq 100$ Oe is applied. Thereafter, the growth of the
magnetization with time [$M(t)$] was recorded. The resulting $M(t)$ curves are
plotted in Fig. 20. The time evolution of magnetization curves are found to
follow a stretched exponential behaviour specified by the Kohlrausch-Williams-
Watts relationKroder  _et al._ (2019); Alvarez  _et al._ (1991)

$M(t)=M_{\rm 0}-M_{\rm g}~{}{\rm exp}[-(t/\tau)^{\beta}].$ (32)

Here, $M_{\rm 0}$ is the intrinsic magnetization at $t=0$, $M_{\rm g}$ is
associated with the glassy component of the magnetization, $\tau$ is the
characteristic relaxation time, and $\beta$ is the stretching exponent.
Typically, the value of $\beta$ varies between 0 and 1 which decides the
dynamics of a spin system. $\beta$ is also a function of temperature and is
strongly dependent on the nature of energy barrier involved in the relaxation
process. Following Eq. (32), when $\beta=0$, $M(t)=$ constant; means no
relaxation. Similarly, $\beta=1$ implies relaxation of a spin system with a
single time constant due to the presence of uniform energy barrier. On the
other hand, $\beta<$ 1 implies the presence of the distribution of non-uniform
energy barriers, typically observed for SG and superparamagnetic systems.Bag
_et al._ (2018); Kroder  _et al._ (2019); De  _et al._ (2012); Tsoi  _et al._
(2005)

The $M(t)$ curves for $T=70$ K, 10 K, and 2 K are well fitted by Eq. (32),
yielding $\beta\simeq 0.48$, 0.38, and 0.37, respectively. These values are
less than 1 suggesting the evolution of magnetization through a number of
intermediate metastable states. The value of $\tau$ increases with decreasing
temperature as expected for glassy systems.Bag  _et al._ (2018); Li  _et al._
(1998) The relaxation behaviour observed at $T=10$ K and $T=2$ K is quite
natural because both of these temperatures are well below $T_{\rm f2}$.
However, the slow relaxation behaviour observed at high temperatures ($T=70$ K
$>$ $T_{\rm f2}$) is unusual and suggests that the persistence of
SG/superparamagnetism beyond $T_{\rm f2}$ in the ferrimagnetically ordered
state.

####  III.10.2 Magnetic Memory Effect

Figure 21:  Memory effect measured as a function of temperature in FC (upper
panel) and ZFC (lower panel) protocols in $H=100$ Oe, as described in the
text. Inset of the upper panel magnifies the features at the interruption
points: $T=5$ K, 12 K, and 70 K. The difference in magnetization, $\Delta
M(=M^{\rm mem}_{\rm ZFCW}-M^{\rm ref}_{\rm ZFCW})$ vs $T$ is plotted in the
inset of the lower panel.

Since the bifurcation of the DC ZFC and FC $\chi(T)$ data occurs for both SG
and superparamagnetic systems, the history-dependent measurements are required
in order to delineate the microscopic character of the spin system. Therefore,
in the following, we have described the magnetic memory effect measurements in
both FC and ZFC protocols.

In the FC protocol, the sample was cooled in a small magnetic field of $H=100$
Oe from high temperature paramagnetic state ($T\geq 200$ K) to the lowest
measured temperature $T=2$ K at a constant rate of 0.5 K/min with intermediate
stops at three different temperatures ($T_{1}^{\rm w}=70$ K, $T_{2}^{\rm
w}=12$ K, and $T_{3}^{\rm w}=5$ K). At each stop, the magnetic field was
switched off and a waiting time of $t_{\rm w}=3$ hours was given for the
magnetization to relax. After each $t_{\rm w}$, the same magnetic field was
applied and field cooled cooling (FCC) process was resumed. In this way, the
recorded magnetization ($M^{\rm w}_{\rm FCC}$) is plotted in the upper panel
of Fig. 21 which shows step like features at each stopping temperature. Once
it reached 2 K, the magnetization was recorded by heating the sample at the
same rate (0.5 K/min) back to 200 K in the same magnetic field without any
intermediate stop. The resulting field cooled warming magnetization data are
referred as $M^{\rm mem}_{\rm FCW}$ in the upper panel of Fig. 21. As depicted
in the inset of the upper panel of Fig. 21, $M^{\rm mem}_{\rm FCW}$ also
exhibits change of slope at each stopping temperature ($T_{2}^{\rm w}=12$ K
and $T_{3}^{\rm w}=5$ K) as in $M^{\rm w}_{\rm FCC}$. These characteristic
features clearly imply that the system tries to remember the thermal history
of magnetization during cooling, thus, showing the magnetic memory. A weak
change of slope was also seen at the stopping temperature, $T_{1}^{\rm w}=70$
K which is well above $T_{\rm f2}$. Similar behaviour is also reported for
reentrant SG compound Lu2MnNiO6 and it is mentioned that the unusual behaviour
where memory exists above $T_{\rm f}$ and below $T_{\rm C}$ can be attributed
to the effect of magnetic frustration due to competing FM and AFM
interactions.Manna  _et al._ (2015) A FCC curve, $M^{\rm ref}_{\rm FCC}$ in
the same field without any interruption is also measured as a reference.

In the ZFC protocol, the sample was cooled in zero magnetic field form $T\geq
200$ K to 2 K at a constant rate of 0.5 K/min with three intermediate stops at
$T_{1}^{\rm w}=70$ K, $T_{2}^{\rm w}=12$ K, and $T_{3}^{\rm w}=5$ K. At each
stop, the sample was allowed to relax for a waiting time of 3 hours. After
reaching 2 K, the magnetization $M^{\rm mem}_{\rm ZFCW}$ was collected by
warming the sample upto 200 K, after applying a small magnetic field of 100
Oe. The reference data, $M^{\rm ref}_{\rm ZFCW}$ were also taken by measuring
the magnetization during warming in the same magnetic field 100 Oe, after the
sample was cooled in zero magnetic field without any intermediate stops. From
the data presented in the lower panel of Fig. 21, it can be seen that there is
neither any clear dip nor any change in slope in the $M^{\rm mem}_{\rm ZFCW}$
data at the stopping temperatures, implying the absence of ZFC memory effect.
The difference in magnetization, $\Delta M(=M^{\rm mem}_{\rm ZFCW}-M^{\rm
ref}_{\rm ZFCW})$ vs $T$ is also plotted in the inset of the lower panel of
Fig. 21 to highlight no memory effect. In order to make sure that there is no
ZFC memory, we have also studied memory effect by measuring AC susceptibility
in the ZFC protocol, following the same procedure as discussed above. Similar
to the DC $\chi(T)$, the real ($\chi^{\prime}$) and imaginary
($\chi^{\prime\prime}$) parts of the AC susceptibility data (not shown) don’t
show any change of slope at the stopping temperatures. This further proves
that no memory is imprinted by aging under zero field.

The above memory effect can be understood from the simple two-state model
proposed by Sasaki $et.~{}al.$Sasaki  _et al._ (2005) and Tsoi $et.~{}al.$Tsoi
_et al._ (2005) for non-interacting magnetic nano particles
(superparamagnets). In this model, it is assumed that a superspin associated
with the dipole magnetic moment of a nano particle can occupy one of the two
states with energies $-KV\pm HM_{\rm s}V$, where $K$ is the bulk anisotropy
constant, $V$ is the volume of the nanoparticle, and $H$ is the applied field.
Therefore, a broad distribution of particle volumes results in a broad
distribution of anisotropic energy barriers. The occupation probability
$p_{1}(t)$ of one of the two states, in which the superspin is antiparallel or
parallel to the applied field is $p_{1}(t)=0.5$ [i.e. $M(t)=0$] at any time
$t$, if $p_{1}(t=0)=0.5$ and $H=0$. Thus, in a ZFC process, which starts from
a initial demagnetized state [$M(t=0)=0$], $p_{1}(t)$ and hence the total
magnetization is independent of the waiting time ($t_{w}$), whereas for a FC
process which starts from a initial magnetized state, $p_{1}(t)$ and hence the
total magnetization is dependent on $t_{w}$. In the light of this model, a
superparamagnet should not show ZFC memory but it can show FC memory simply
because of the blocked (frozen) superspins. On the other hand, a SG system can
show both FC and ZFC memories which are well explained by Sasaki
$et.~{}al.$Sasaki  _et al._ (2005) considering the random energy modelDerrida
(1981); *Bouchaud1705 as well as the droplet theoryFisher and Huse (1988)
proposed for SG systems. The above models have been employed to describe the
experimental data of various superparamagnets and SG systems.Sasaki  _et al._
(2005); Tsoi  _et al._ (2005); Bandyopadhyay and Dattagupta (2006); Chen  _et
al._ (2005) Thus, the absence of ZFC memory discriminates the dynamics of LNMO
from the behaviour of a SG and establishes the superparamagnetic nature at low
temperatures.Tsoi  _et al._ (2005); De  _et al._ (2012) It is quite surprising
that despite having large average particle sizes ($\sim 100$ nm), the compound
still behaves like a superparamagnet. It is to be noted that the
unconventional superparamagnetism is also reported in several compounds in
polycrystalline form.Bajpai and Banerjee (2000); *Ba1anda224421

####  III.10.3 Memory Effect using Magnetic Relaxation

Figure 22:  Magnetic relaxation measured in negative $T$-cycle for (a) ZFC and
(b) FC methods, as described in the text. Insets: Magnetic relaxation at 12 K
measured during $t_{1}$ and $t_{3}$ for negative $T$-cycle and in ZFC and FC
methods. The solid lines are the fits using Eq. (32). Magnetic relaxation data
measured in positive $T$-cycle for ZFC and FC methods are shown in (c) and
(d), respectively.

To investigate the memory effect in further detail, we have performed the
magnetic relaxation measurements following the protocol reported by Sun
$et.~{}al.$Sun  _et al._ (2003) for both negative and positive-$T$ cycles.

$Negative~{}T-cycle$: In a negative temperature cycle, we have measured the
magnetic relaxation in both ZFC and FC protocols and are plotted in Fig. 22(a)
and (b), respectively. In the ZFC procedure, the sample was first cooled from
200 K down to 12 K in zero field. At 12 K, a small field of 100 Oe was applied
and $M(t)$ was recorded for $t_{1}=2$ hours, which is found to grow
exponentially with $t$. The sample was again cooled down to 5 K in the same
magnetic field and $M(t)$ was measured for $t_{2}=2$ hours. The nature of
$M(t)$ curve was found to be almost constant with $t$. Thereafter, the
temperature was restored back to 12 K and $M(t)$ was recorded for $t_{3}=2$
hours in the same field. At this temperature, the $M(t)$ curve was again found
to grow exponentially with $t$. As shown in the inset of Fig. 22(a), the
$M(T)$ data measured during $t_{1}$ and $t_{3}$ follow a continuous growth
curve. In the FC process, the sample was cooled in a small magnetic field of
100 Oe down to 12 K. Once it reached 12 K, the magnetic field was switched off
and the decay of magnetization with $t$ was measured for $t_{1}=2$ hours. The
sample was further cooled down to 5 K in zero field and $M(t)$ was recorded
for $t_{2}=2$ hours at 5 K. This $M(t)$ curve was found to be almost constant
with $t$. Subsequently, the sample was heated back to 12 K in zero field and
$M(t)$ was recorded for $t_{3}=2$ hours at 12 K. The $M(t)$ curve measured
during $t_{3}$ was found to decay exponentially with $t$ as a continuation of
$M(t)$ curve recorded during $t_{1}$ [see the inset of Fig. 22(b)].

Thus, the continuous growth and decay of magnetization during $t_{1}$ and
$t_{3}$ obtained for the ZFC and FC processes, respectively indicate that the
state of the system at 12 K is recovered after a temporary cooling. This a
clear demonstration of the memory effect where the system tries to remember
the initial state even after going through a change of magnetization. These
continuous curves could be fitted well by the stretched exponential function
[Eq. (32)] with $\beta\simeq 0.37$ and 0.48 for the ZFC and FC processes,
respectively which are consistent with the magnetic relaxation measurements.

$Positive~{}T-cycle$: Similar to the negative $T$-cycle, we have also done
magnetic relaxation measurements in positive $T$-cycle in both ZFC and FC
protocols. For ZFC procedure, the sample was first cooled from 200 K down to 5
K in zero magnetic field and $M(t)$ was recorded for $t_{1}=2$ hours, after
applying a small magnetic field of 100 Oe. Then, the sample was heated up to
12 K in the same field and $M(t)$ was measured for $t_{2}=2$ hours.
Thereafter, the sample was again cooled back to 5 K in the same field and
$M(t)$ was recorded for $t_{3}=2$ hours. In the FC procedure, the sample was
first cooled down to 5 K in a small magnetic field of 100 Oe. Once the
temperature reached 5 K, the magnetic field was switched off and $M(t)$ was
recorded for $t_{1}=2$ hours. By keeping the field zero, the sample was heated
to 12 K and $M(t)$ was recorded for $t_{2}=2$ hours. In zero field, the sample
was further brought back to 5 K and $M(t)$ was measured for $t_{3}=2$ hours.
The measured ZFC and FC data are presented in Fig. 22(c) and (d),
respectively. Unlike the negative $T$-cycle there is no continuity found in
the $M(t)$ data measured during $t_{1}$ and $t_{3}$ for both ZFC and FC
measurements in the positive $T$-cycle. This clearly suggests that positive
$T$-cycling or temporary heating erases the memory and re-initializes the
relaxation process in both ZFC and FC methods. Thus, no memory effect is
observed when the temperature is restored back to 5 K.

The asymmetric response of magnetic relaxation with respect to both negative
and positive $T$-cycles is typically observed for both SG and
superparamagnetic systems. In SG systems, this behaviour can be explained on
the basis of the hierarchical model.Lefloch  _et al._ (1992); Sun  _et al._
(2003) Similarly, for superparamagnetic systems, this behaviour is very well
explained by Tsoi $et.~{}al.$Tsoi  _et al._ (2005) and Bandyopadhyay
$et.~{}al.$Bandyopadhyay and Dattagupta (2006) in the light of simple two-
state superparamagnetic modelSasaki  _et al._ (2005). Thus, the observed
memory effects in ZFC and FC protocols for the negative $T$-cycle further
justifies the superparamagnetic nature of the system under investigation.

##  IV Summary

We have successfully synthesized the polycrystalline sample of a new modified
cubic spinel compound LNMO. It is found to crystallize in a cubic structure
with a non-centrosymmetric space group $P4_{3}32$ and exhibits 1:3 cation
order of Ni2+ and Mn4+ ions. Physical property measurements suggest
semiconducting nature of the compound which exhibits a long-range
ferrimagnetic ordering at $T_{\rm C}\simeq 125$ K. The analysis of the neutron
diffraction data reveals a collinear ferrimagnetic spin structure, with
magnetic moments aligned along [110] direction. The moments of each Ni2+ or
Mn4+ sublattice are coupled ferromagnetically whereas the inter-sublattice
interaction is antiferromagnetic. The compound is still frustrated due to
competing AFM and FM interactions which is also the reason for the reduction
of frustration ratio ($f$), despite a highly frustrated pyrochlore lattice
geometry. The reduced ordered moment of Ni2+ or Mn4+ ions also indicates the
presence of significant frustration.

The critical exponents obtained from the analysis of magnetization data near
$T_{\rm C}$ via modified Arrott and Kouvel-Fisher plots fall in the category
of 3D XY universality class. The reliability of these critical exponents and
the value of $T_{\rm C}$ are further confirmed from the scaling of
magnetization isotherms. A reversible MCE with a large value of $\Delta S_{m}$
and $RCP$ has been observed over a wide temperature range across $T_{\rm C}$
which can be ascribed to the effect of magnetic frustration. Even the critical
analysis of field dependent $\lvert\Delta S_{\rm m}^{\rm peak}\lvert$ and
$RCP$ curves produce exponents close to the ones obtained from the critical
analysis of magnetization. The second order nature of the phase transition has
also been confirmed from the universal scaling of $\Delta S_{\rm m}$ and the
nature of $n(H,T)$ curve. Our results demonstrate that, LNMO is a promising
refrigerant material, where frustration associated with the competing exchange
interactions drives MCE.

The magnetic relaxation below $T_{\rm b}$ follows stretched exponential
function demonstrating that the system evolves through a number of metastable
states. $\chi^{{}^{\prime\prime}}(T)$ below $T_{\rm C}$ depicts multiple
anomalies, likely due to the effect of magnetic frustration. The hump at
$T=T_{\rm f2}$ shows strong frequency dependency with unusual dynamics. It
follows a Arrhenius behaviour in the high temperature regime ($T>27$ K),
suggesting superparamagnetic behaviour and a power law behaviour in the low
temperature regime ($T<27$ K), suggesting SG dynamics at low temperatures.
However, the absence of ZFC memory effect below $T_{\rm b}\simeq 120$ K rules
out the possibility of SG transition and confirms the superparamagnetic
behaviour down to the lowest measured temperature. Indeed, the behaviour of DC
$\chi(T)$ measured in FC and ZFC conditions also seems to substantiate the
superparamagnetic nature of the compound with blocking temperature $T_{\rm
b}\simeq 120$ K. Nevertheless, the critical slowing down behaviour following a
power law at low temperature and the absence of ZFC memory contradict each
other and warrants further investigations.

##  V acknowledgments

We would like to acknowledge SERB, India bearing sanction order No.
CRG/2019/000960 and BRNS, India bearing sanction Grant No.
37(3)/14/26/2017-BRNS for financial support.

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