# Experimental and theoretical demonstration of negative magnetization induced
by particle size reduction in nano-form Gd1-xCaxMnO3

Papri Dasgupta1 Sanjukta Paul${}^{{}^{3}}$ S. Kumar1 Sudhakar Yarlagadda2
Chandan Mazumdar2 1Physics Division, Jadavpur University, Kolkata, 700032,
India  2CMP Division, Saha Institute of Nuclear Physics, HBNI, Kolkata 700064,
India  3School of Physical Sciences, National Institute of Science Education
and Research, Jatni 752050, India

###### Abstract

We report a pervasive phenomenon of gradual emergence of negative
magnetization in typical $R_{1-x}D_{x}$MnO3 orthorhombic perovskite manganites
in nano form (with $R^{3+}$ being a trivalent high-magnetic-moment ion from
the heavier rare-earth elements Gd, Tb, Dy and $D^{2+}$ refers to divalent
alkaline elements Sr, Ca, etc.), while the bulk form manifests no negative
magnetization. Extensive magnetization studies have been carried out in
Gd1-xCaxMnO3 manganites around half doping. We demonstrate experimentally that
particle size reduction in nano-form manganites enhances the propensity of the
system to exhibit negative magnetization and provide a theoretical explanation
for the phenomenon. To test the universality of our findings, we have extended
the measurement to Dy0.6Ca0.4MnO3 and obtained similar results.

##  I Introduction

Over the past few decades, complex oxides such as the manganites have
attracted intense research interest. They display a fascinating tapestry of
charge, spin, and orbital orders when a trivalent rare-earth element is
replaced by a divalent alkaline element cnr ; khomskii0 ; 9 ; hotta ; tokura ;
cheong ; raveau . The rare-earth manganites $R{\rm MnO_{3}}$, for the
compounds with rare-earth element $R$ = La–Dy, exhibit an orthorhombic
perovskite structure; on the other hand, compounds with $R$ = Ho–Lu form in
hexagonal structure pauthenet ; kimura .

A large amount of research has been focused on features of pure and doped
$R{\rm MnO_{3}}$ when $R$ is a light rare-earth element with weak magnetic
moment and large ionic radius. On the other hand insufficient attention has
been paid to pure and doped perovskite manganites involving heavy rare-earth
elements (with large effective moment and small ionic size shannon ) such as
$R{\rm=Gd,~{}Tb,~{}and~{}Dy}$ and divalent alkaline dopant elements Sr, Ca,
etc. Due to the usage of small-sized elements (${\rm Gd,~{}Tb,~{}and~{}Dy}$),
a few important modifications have to be taken in account—the tolerance factor
decreases and the buckling and tilting of the ideal cubic structure becomes
pronounced. Additionally, interesting co-operative effects manifest due to the
large magnetic moment of the ions in the rare-earth network—namely, a reversal
of the magnetization results due to the coupling of the two interacting
magnetic networks yusuf ; biswas . Furthermore, these manganites, $R{\rm
MnO_{3}}$ ($R{\rm=Gd,~{}Tb,~{}and~{}Dy}$), display multiferroicity with
magnetic order being accompanied by electric polarization kimura2 ; tokura1 ;
kimura3 ; cheong2 ; arima .

Although permanent magnetic systems generally show positive magnetization
values, there are some systems (having two sublattices) which may exhibit
negative magnetization due to the negative-exchange coupling between the two
ferromagnetically ordered sublattices yusuf . However, two individually
ferromagnetic sublattices are not always needed for the manifestation of
negative magnetization. There exists another interesting set of compounds,
that also display negative magnetization, wherein a disordered paramagnetic
sublattice is coupled with an ordered ferromagnetic subtlattce via negative-
exchange interaction. In these compounds having two sublattices A and B,
sublattice A orders along the external field below a magnetic ordering
temperature $T_{\rm C}$ and influences the sublattice B (which remains
paramagnetic at the measuring temperature $T\geq T_{\rm C}$) to align opposite
to the field of sublattice A. Below a certain temperature called the
compensation temperature (which is sizeably smaller than $T_{\rm C}$), if the
moment of sublattice B becomes larger than that of A, the net magnetization
becomes negative yusuf ; biswas . A finite magnetic anisotropy should also be
present in the system to prevent the rotation of the net magnetic moment,
below the compensation temperature, in the direction of applied magnetic
field. Some examples of such compounds are orthorhombic perovskite manganites
$R_{1-x}D_{x}$MnO3 with $R^{3+}$ being Gd, Tb, and Dy and $D^{2+}$ referring
to Sr or Ca yusuf ; biswas ; 4 ; 5 ; snyder ; ma ; yingnan ; nagaraja ; jin ;
elizabeth .

In recent years, research on compounds exhibiting negative magnetization has
been gaining momentum as these materials can be utilized in digital data
storage systems by exploiting their ability to reversibly switch between two
distinct magnetization states (i.e., positive and negative) when the
temperature or an externally applied magnetic field or light is varied yusuf ;
7 . Bipolar switching of the magnetization can be employed to make
magnetoelectronic devices such as volatile memories. Furthermore, the low cost
of synthesis of manganites by sol-gel technique makes them suitable for large-
scale industrial application 8 .

Generally, among the perovskite manganites, negative magnetization can be
obtained in a low-bandwidth material having mixed-valent Mn ions (i.e., Mn3+
and Mn4+). Low bandwidth can be realized in systems containing the heavier
rare-earth elements (Gd–Dy) biswas ; mixed valency of Mn ions is obtained by
partially substituting trivalent rare-earth ions by divalent alkaline-earth
ions. These undoped heavier rare-earth manganites, including GdMnO3 (whose
doped version is studied here), have been reported to exhibit
antiferromagnetic (AFM) insulating behavior tokura1 ; 4 . We have chosen Gd-
based manganites because of the simplicity resulting from the zero orbital
angular momentum in Gd3+ ions. With the substitution of divalent ions (e.g.,
Ca2+), an equivalent number of Mn4+ ions are introduced in the Mn sublattice
of Gd1-xCaxMnO3 to make the system charge neutral. With the Gd ions remaining
paramagnetic down to low temperatures, the Mn-Mn interaction leads to
different ordered states in Gd1-xCaxMnO3 depending on the relative percentage
of Mn3+ and Mn4+ ions. The phase diagram of Gd1-xCaxMnO3, obtained from
magnetoresistive measurements, report ferromagnetic insulating phase for
$x<0.5$, which transforms into charge ordered (CO) state with AFM interactions
for $x\geq 0.5$ 4 ; 10 . In the low-doped region ($0<x<0.5$), the
ferromagnetically ordered Mn-spins force the Gd-spins to align opposite to its
direction, i.e., opposite to $H_{\rm ext}$, due to a weak negative exchange
interaction between them 4 ; thus, a ferrimagnet results at very low
temperatures. With decreasing temperature, as the contribution of the magnetic
moment of paramagnetic Gd ions keeps on increasing in the direction opposite
to the external magnetic field, the net moment decreases and can become
negative. At higher doping concentrations ($x>0.5$), the AFM domains become
larger in extent and percolate the sample causing the total magnetization to
increase steadily with decreasing temperature 10 .

Although the phenomenon of negative magnetization has been studied in
manganite compounds in bulk form yusuf ; 4 ; 10 ; 11  and in thin-film form
paturi2 , the role of particle size in nano-form manganites has not been
investigated. In the nano systems surface effects in nanoparticles can alter
the magnetic behavior. In this work, we carried out a detailed study of
Gd1-xCaxMnO3 compounds around half-doping (i.e., at x = 0.4, 0.5, and 0.6) in
their bulk and nanoparticle form (under the influence of external magnetic
fields of various strengths) in order to understand the effect of particle
size on negative magnetization. While the bulk forms of Gd1-xCaxMnO3 (at
$x=0.4~{}{\rm and}~{}0.5$) exhibit charge ordering and no signature of
negative magnetization, its nanoparticle forms do exhibit negative
magnetization at low temperatures although they remain devoid of any charge
ordering. However, when temperature is varied, both bulk and nano forms
exhibit a hump in magnetization for $x=0.4~{}{\rm and}~{}0.5$; this is caused
by ferromagnetic domains of Mn-spin sublattice aligning at higher temperatures
followed by Gd spins in this region aligning antiparallel to the Mn spins at
lower temperatures due to a weak antiferromagnetic Mn-Gd coupling.
Furthermore, in the antiferromagnetic domains of Mn-spin sublattice at any
doping, the localized Gd spins in this region (effectively decoupled from the
antiferromagnetic Mn sublattice) produce a paramagnetic response to the weak
external magnetic field. In nanoparticles, owing to the decrease in rigidity
of the lattice at the surface, the concentration of Gd ions (Ca ions) is
higher at the surface (core) because they produce lower (higher) tolerance
factor. Higher than the stoichiometric Gd concentration at $x=0.4~{}{\rm
and}~{}0.5$ yields negative magnetization similar to Gd1-xCaxMnO3 at lower
doping (i.e., lower than $x=0.4~{}{\rm and}~{}0.5$, respectively) in the
bulk/thin-film form 4 ; paturi2 . Additionally, the magnetization is also
significantly enhanced due to double exchange in the outer shell of the
nanoparticles.

The rest of the paper is organized as follows. In Sec. II, we describe both
the preparation methods, of bulk and nanoparticle samples of Gd1-xCaxMnO3, as
well as the measurements performed. Then, in Sec. III, we provide the
theoretical framework employed to analyze the observed magnetization and
charge ordering. Next, in Sec. IV, we present our data showing the increase in
propensity for negative magnetization as the particle size decreases in nano-
form samples at various dopings. We also offer our theoretical explanation for
this phenomenon. We close in Sec. V with our concluding observations and
discuss possible future directions.

##  II Experimental details

Polycrystalline samples of Gd1-xCaxMnO3 (x = 0.4, 0.5, 0.6) have been prepared
by well known sol-gel technique. For the preparation of Gd1-xCaxMnO3, Gd2O3,
CaCO3 and MnO2 were used as starting materials. Appropriate amount of oxides
were separately dissolved in a HNO3 solution (Oxalic acid is also added in
case of MnO2). These solutions were mixed, and an amount of citric acid
equivalent to the total number of moles of metal ions was added under moderate
heating and stirring conditions. Subsequently, mixture was slowly evaporated
at 80oC–90oC in water bath resulting in the formation of gel which was heated
to 250oC to remove the organic matter and decompose the nitrates of the gel.
The black ash thus obtained was ground and further heated to 550oC for 5 hours
to kick-off the remaining organic matter and then pelletized. The pellets then
finally sintered for 24 hours at 1250oC to obtain bulk samples. Gd1-xCaxMnO3
nanoparticles of different sizes have been prepared by subsequently sintering
the pellets at 700oC, 800oC, 900oC, 1000oC and 1100oC for 3 hours. In the rest
of this paper, the corresponding nanoparticle forms are identified by
mentioning the sintering temperature after the compound, for example
Gd0.5Ca0.5MnO3 annealed at 700oC is written as Gd1-xCaxMnO3-700. The bulk
samples are represented in the manuscript as Gd1-xCaxMnO3-bulk. X-ray
diffraction (XRD) study was carried out at room temperature with TTRAX-III
diffractometer (M/s Rigaku, Japan) using Cu-K$\alpha$ source having wavelength
1.5406 Å. Lattice parameters have been estimated from XRD pattern using the
FullProf software 11a . Magnetization was measured using VSM-SQUID
magnetometer (M/S Quantum Design Inc, USA). Temperature dependence of
magnetization in zero-field-cooled (ZFC) and field-cooled (FC) protocol in the
warming cycle has been carried out under different magnetic fields (100 Oe $<$
H $<$ 70,000 Oe) in the temperature range 5 K - 330 K. Field dependence of
magnetization (M-H) and magnetic relaxation measurement, etc., has also been
carried out.

Figure 1: (colour online) Room temperature X-ray diffraction pattern of bulk
${\rm Gd_{1-x}Ca_{x}MnO_{3}}$ (x = 0.4, 0.5, 0.6) compounds. The experimental
data (red points), the calculated (black line) XRD patterns, allowed Bragg
position by the space group (green lines), and the difference of experimental
data and calculated values (blue lines) for all the compounds are displayed.

Room temperature X-ray diffraction pattern of polycrystalline bulk samples of
Gd1-xCaxMnO3 (x = 0.4, 0.5, 0.6), prepared by sol-gel technique, is shown in
Fig. 1. All the XRD lines confirm to Pnma space group suggesting the materials
formed in single phase within the limit of resolution of our diffraction
experiment.

##  III Theoretical framework

We will analyze magnetization in ${\rm Gd_{1-x}Ca_{x}MnO_{3}}$ (GCMO)
structures (bulk or nanomaterials) using ZFC, FC, and isothermal hysteresis
curves. We first present the following Hamiltonian, involving a single band
for simplicity, without sacrificing any essential physics:

$\displaystyle H=H_{\rm KE}+H_{\rm pol}+H_{\rm M}.$ (1)

The kinetic energy term $H_{\rm KE}$ is given by

$\displaystyle H_{\rm KE}=-te^{-\alpha E_{p}/\hbar\omega}\sum\limits_{\langle
i,j\rangle}\left[{\cos\left({\theta_{i,j}\over
2}\right){c^{\dagger}}_{i}c_{j}}+{\rm H.c.}\right],$ (2)

where $t$ is the fermion hopping amplitude that is attenuated by the electron-
phonon coupling $(E_{p}/\hbar\omega)^{1/2}$ with $E_{p}$ being the polaron
energy, $\omega$ the optical phonon frequency, and $\alpha$ a constant of the
order unity; $c_{j}$ is the $e_{\rm g}$ electron destruction operator;
$\theta_{i,j}$ is the angle between $S=3/2$ spins (originating from localized
$t_{\rm 2g}$ spins) at ${\rm Mn}$ sites $i$ and $j$; and
$\cos(\theta_{i,j}/2)$ provides the modulation due to large Hund’s coupling
between the spin of the hopping $e_{\rm g}$ electron and the spins of the
localized $t_{\rm 2g}$ electrons gennes ; Izyumov . Now, the small size of the
${\rm Gd^{3+}}$ ion (with the size being smaller than the space created by the
surrounding ${\rm MnO_{6}}$ octahedra, i.e., with the tolerance factor being
small) induces buckling of the ${\rm Mn-O-Mn}$ bonds leading to a small
hopping $t$; since the electron-phonon coupling is strong, the kinetic energy
is small. Then, in the presence of disorder (such as cation disorder), even
when the disorder is weak, this leads to localization (in fact, site
localization) of electrons hwang .

The second term $H_{\rm pol}$ in Eq. (1), is the polaronic Hamiltonian given
by

$\displaystyle H_{\rm pol}=-\sum_{j,\delta}\left[\beta
E_{p}+\frac{\left[t_{j,j+\delta}\cos(\theta_{j,j+\delta}/2)\right]^{2}}{2\gamma
E_{p}}\right]n_{j}(1-n_{j+\delta}),$ (3)

where the first coefficient $\beta E_{p}$ ($\beta$ being of the order unity)
is due to cooperative electron-phonon interaction and portrays nearest-
neighbor (NN) electron-electron repulsion due to incompatible distortions of
NN oxygen cages surrounding occupied ${\rm Mn}$ ions. The value of $\beta$
depends on the nature of compatibility in the orbital order [see Ref. ME, ].
Next, the second coefficient
$\left[t_{j,j+\delta}\cos(\theta_{j,j+\delta}/2)\right]^{2}/(2\gamma E_{p})$
results from second-order perturbation theory and involves virtual processes
of hopping of a fermion from an ${\rm Mn}$ site to the NN ${\rm Mn}$ site and
back and is present even when we consider the simpler Holstein model sdadsy1 .
The factor $\gamma$ is of the order unity and increases with the increase in
occupancy of the neighbors of the ${\rm Mn}$ site $j+\delta$ (see Fig. 2);
this is because of the NN repulsion felt by the fermion when it is virtually
present at site $j+\delta$ in the intermediate state of the second-order
process sp . Thus, the second coefficient in Eq. (3) shows that not only is NN
${\rm Mn}$ site occupation discouraged, the next-nearest-neighbor ${\rm Mn}$
site occupation is also discouraged although to a weaker extent. Furthermore,
although $n_{j}$ is the total number of electrons in both the $e_{\rm g}$
orbitals at the ${\rm Mn}$ site $j$, it can only take a maximum value of 1 due
to strong on-site electron-electron repulsion and strong Hund’s coupling. It
is important to note that, even though the electrons are localized, the
electrons can virtually hop to the neighboring site and produce NN
ferromagnetic spin alignment as well as nearest-neighbor, next-nearest-
neighbor, and next-to-next-nearest-neighbor charge repulsion through the
second coefficient
$\left[t_{j,j+\delta}\cos(\theta_{j,j+\delta}/2)\right]^{2}/(2\gamma E_{p})$
which is much smaller than the first coefficient $\beta E_{p}$. Thus the
charge-order energy scale set by the first coefficient $\beta E_{p}$ is higher
than the ferromagnetic spin-order energy scale set by the second coefficient
$\left[t_{j,j+\delta}\cos(\theta_{j,j+\delta}/2)\right]^{2}/(2\gamma E_{p})$.

Figure 2: (colour online) Hopping process, corresponding to second-order
perturbation theory, shown for a two-dimensional Mn lattice. Schematic
representation of a fermion, originating at Mn site $j$, hopping to its NN Mn
site $j+\delta$ (the intermediate site) and coming back. Here empty
continuous-line circle corresponds to empty site, while continuous-line circle
with small dot indicates fermion position. The intermediate site $j+\delta$
has each of its three NN Mn sites (depicted by dashed-line circles) either
occupied by a fermion or empty. Semicircle at site $j$ represents full
distortion of the lattice ions at that site with corresponding energy
$-E_{p}(+E_{p})$ if the fermion is present (absent) at that site.

The next term $H_{\rm M}$ in Eq. (1) pertains to the magnetic interactions; in
$\rm LaMnO_{3}$ it refers to the superexchange anderson  interactions which
generate A-AFM order. In Gd1-xCaxMnO3, it is given by

$\displaystyle H_{\rm M}=\sum\limits_{\langle
i,j\rangle}J_{i,j}\vec{S}_{i}\cdot\vec{S}_{j}+J_{\rm Mn-
Gd}\sum\limits_{\langle i,\eta\rangle}\vec{S}_{i}\cdot\vec{\cal{S}}_{\eta},$
(4)

where $J_{i,j}$ is the ${\rm Mn-Mn}$ superexchange coupling between the spins
on the nearest neighbors and $J_{\rm Mn-Gd}$ ($<|J_{i,j}|$) is a weaker
antiferromagnetic coupling between a ${\rm Gd^{3+}}$ ion at the center of the
unit cell (at location $\eta$) and its eight nearest-neighbor ${\rm Mn}$ ions
at the corners of the unit cell snyder . In the above expression, the
magnitude of the superexchange coefficient $J_{i,j}$ is much smaller than the
second coefficient
$\left[t_{j,j+\delta}\cos(\theta_{j,j+\delta}/2)\right]^{2}/(2\gamma E_{p})$
in Eq. (3). At low temperatures, the antiferromagnetic coupling $J_{\rm Mn-
Gd}$ can produce a metastable state with negative magnetization (actually a
ferrimagnetic state) when at higher temperatures a ferromagnetic state
results. The coupling between the rare-earth network and the manganese network
is a feature absent in La1-xCax MnO3.

##  IV Results and discussions

Figure 3: (colour online) Temperature dependent magnetization of
Gd1-xCaxMnO3-bulk ($x$ = 0.4, 0.5, 0.6) at H = 100 Oe. FC (ZFC) data is
plotted in red (black). Inset shows the enlarged view of the magnetization
curve (left axis) to point out the CO temperature and the corresponding peak
in the derivative of magnetization curve (right axis). For $x=0.6$ case,
theoretical response for free Gd3+ ions (plotted in blue below 50 K) shows
their dominant contribution to the total magnetization at lower temperatures.
Figure 4: (colour online) Inverse magnetic susceptibility of bulk Gd1-xCaxMnO3
($x$ = 0.4, 0.5, 0.6) measured at 100 Oe under ZFC conditions. The linear
regions, indicating Curie-Weiss behavior, are depicted by solid lines. In the
depicted linear regions, ZFC and FC curves coincide as shown in Fig. 3.

In this paper we analyze Gd1-xCaxMnO3 samples at concentrations in the
neighborhood of half-doping, i.e., for $x=0.4$, $0.5$, and $0.6$. Now, based
on the phase diagram for bulk samples (see Ref. 10, ), for $x<0.5$ we have a
ferromagnetic insulator, whereas for $0.5\leq x\leq 0.8$ we get an
antiferromagnetic insulator at lower temperatures. For $x<0.5$, based on Ref.
sp, , ferromagnetic insulator can be explained as follows. Since cooperative
electron-phonon interaction is strong, a NN electron-hole pair has a
ferromagnetic interaction
$\left[t_{j,j+\delta}\cos(\theta_{j,j+\delta}/2)\right]^{2}/(2\gamma E_{p})$
[as shown in Eq. (3)] which is much stronger than other magnetic interactions
($|J_{i,j}|$ and $J_{\rm Mn-Gd}$). Hence, a ferromagnetic cluster (magnetic
polaron) is generated in the vicinity of a hole. In fact, the hole (through
virtual hopping) will polarize electrons that are nearest-neighbor as well as
those that are next-nearest-neighbor and next-to-nearest-nearest-neighbor to
form a magnetic polaron. A collection of interacting magnetic polarons will
create a ferromagnetic-insulator region. As regards the doping region $x\geq
0.5$, several theories have been put forth to explain antiferromagnetism
khomskii0 ; hotta ; we do not propose any new explanation beyond these
theories for antiferromagnetism.

We will now provide an understanding of the magnetization displayed in the
ZFC, FC, and hysteresis curves for bulk and nanoparticle forms.

###  IV.1 FC and ZFC cases for weak external field

####  IV.1.1 Bulk samples

We will first analyze the temperature dependent magnetization M(T) curves for
Gd1-xCaxMnO3-bulk ($x$ = 0.4, 0.5, and 0.6), obtained under ZFC and FC
conditions at 100 Oe external field, shown in Fig. 3. Enlarged view of the
M(T) curves below 300 K, shown in insets of Fig. 3, displayed discernible
humps in the temperature range 255 K–275 K. The anomaly can, however, be
prominently manifested on taking the derivative of the M(T) curves (see insets
and their right axes in Fig. 3). This is a commonly observed behavior in many
manganite systems including Gd1-xCaxMnO3 ($x\geq 0.5$) and has been explained
to have arisen due to CO at temperature TCO 13 . It may be noted here that the
signature of CO in our samples has been extended even beyond the reported
region of $x\geq 0.5$ (see Ref. 10, ), as the same signature is observed in
our Gd0.6Ca0.4MnO3 sample [see Fig. 3 bottom (inset)]. Below TCO, in the
temperature range 110 K–200 K, inverse susceptibility [$\chi^{-1}$(T)] curves
of all these compounds (as shown in Fig. 4) follow Curie-Weiss (CW) behavior,
i.e., a linear behavior with $\chi^{-1}({T})=(T-\theta_{\rm CW})/C$. For
Gd0.6Ca0.4MnO3, the values of $\theta_{\rm{CW}}$ and $C$, as estimated from
the CW behavior, are 44.7 K and 9.02 emu K/mole, respectively. The positive
value of $\theta_{\rm{CW}}$ indicates that the dominant exchange interaction
is of the ferromagnetic type. However, indicating antiferromagnetism,
$\theta_{\rm CW}$ is negative for Gd0.5Ca0.5MnO3 and Gd0.4Ca0.6MnO3 (as can be
seen in Fig. 4); $\theta_{\rm{CW}}=-73.6$ K and $C=14.49$ emu K/mole for
Gd0.5Ca0.5MnO3 and $\theta_{\rm{CW}}=-103.5$ K and $C=9.00$ emu K/mole for
Gd0.4Ca0.6MnO3. We note that the magnetic behaviors in our samples of $x=0.4$,
$0.5$, and $0.6$ are in agreement with those reported in Ref. 10, .

Now, the system at $x=0.4$, based on the phase diagram in Ref. 10,  and Fig.
4, is primarily ferromagnetic with small domains that are antiferromagnetic.
Existence of small antiferromagnetic domains at $x=0.4$ is expected due to the
doping proximity to $x=0.5$ where the system is antiferromagnetic; this will
be further justified/clarified below. Hence, for the FC case, at temperatures
${\rm T>T_{\rm Mn-Gd}}$, pertaining to the thermal energy larger than the
magnetic energy for the weak antiferromagnetic coupling $J_{\rm Mn-Gd}$, only
the ferromagnetic fields (generated by the magnetic polarons) are relevant;
here, ferromagnetism decreases with increasing temperatures. At temperatures
below ${\rm T_{\rm Mn-Gd}}$, the antiferromagnetic coupling $J_{\rm Mn-Gd}$
starts becoming relevant and the ${\rm Gd}$ spins (with $S=7/2$) in the
percolating ferromagnetic cluster align antiparallel to the percolating
cluster. At temperatures ${\rm T<T_{\rm Mn-Gd}}$, the ${\rm Gd}$ spins that
are aligned opposite to the percolating ferromagnetic cluster start lowering
the overall magnetism. Contrastingly, the ${\rm Gd}$ spins in the small
antiferromagnetic domains start aligning (as temperature is lowered, i.e., for
$T\lesssim 250$ K) due to the external weak magnetic field (H = 100 Oe) and
contribute to increasing magnetism. This is because, in the small
antiferromagnetic domains, there is no net coupling between ${\rm Gd}$ spins
and the antiferromagnetic lattice of ${\rm Mn}$ spins. Needless to say that,
when the ${\rm Mn}$ spins at the 8 vertices of a unit cell are aligned
antiferromagnetically, the total spin of the ${\rm Mn}$ spins at the 8
vertices is zero; hence, when a ${\rm Gd}$ spin (with $S=7/2$) is introduced
at the center of the cube, the central ${\rm Gd}$ spin has no net coupling to
the total spin of the 8 vertices.

Now, the ZFC curve gives a picture that is qualitatively similar to that of
the FC curve. However, the ZFC curve is further away from equilibrium compared
to the FC curve because its ferromagnetic domains (compared to the FC case)
are less aligned with the external field. This is because the starting state
for the warming cycle of the ZFC case was obtained by cooling at zero field.
Thus, the magnetization peak of the ZFC curve is lower than that of the FC
curve. Consequently, there is a pronounced ZFC-FC bifurcation.

Next, at $x=0.5$, based on the phase diagram in Ref. 10, , the system is
primarily antiferromagnetic with small clusters that are ferromagnetic. In the
FC case, the ferromagnetic clusters are strongly aligned with the external
field and contribute to the magnetism similar to the larger ferromagnetic
regions of the $x=0.4$ case. Again, for the ZFC case, since the small
ferromagnetic clusters are less aligned with the external field (than in the
FC case), we get a smaller magnetization. Since the ferromagnetic domains are
less dominant for the case of $x=0.5$ compared to that of $x=0.4$, the ZFC-FC
bifurcation is less pronounced and the humps are also smaller for $x=0.5$.
Below ${\rm T\lesssim 250}$ K, the ${\rm Gd}$ spins in the large
antiferromagnetic domain start aligning due to the external weak magnetic
field (H = 100 Oe) and contribute to larger magnetism compared to $x=0.4$
case.

We also notice that the ZFC and the FC curves cross between the two
magnetization extrema (i.e., the maximum and the minimum) for x = 0.4 and 0.5.
The ZFC-FC crossover is obtained at Tcross below which the FC M(T) curve
attains values lower than the ZFC M(T) curve. This is because the ZFC curves
have larger AFM domains and smaller ferromagnetic domains (that are less
aligned with the direction of Hext) than the FC case. Furthermore, in the
ferromagnetic domains at $x=0.4$, the negative magnetization due to the ${\rm
Gd}$ spins is more than the positive magnetization due to the ${\rm Mn}$
spins. In fact, based on the formula Gd1-xCaxMnO3, when ${\rm Gd}$ spins are
antiparallel to the ${\rm Mn}$ spins and the weak external magnetic field, we
can work out the effective spin of a unit cell to be
$-\frac{7(1-x)}{2}+2(1-x)+\frac{3x}{2}=-1.5+3x$; here, the axis of
quantization is in the direction of the weak external field. Thus, at very low
temperatures, magnetization in a cluster (containing ferromagnetic Mn
sublattice) can be negative only when $-1.5+3x$ is negative, i.e., $x<0.5$. In
bulk samples at $x=0.4$, as depicted in Fig. 3, it is important to note that
the magnetism due to Gd spins in antiferromagnetic clusters wins over the
negative magnetization contribution from the ferromagnetic domains and keeps
the magnetism positive, i.e, aligned with the external magnetic field.

Lastly, at $x=0.6$ the system is essentially fully antiferromagnetic. Thus,
for both ZFC and FC cases, at lower temperatures only the ${\rm Gd}$ spins get
more aligned due to the external weak magnetic field and magnetization
increases in a similar fashion. Furthermore, as shown in Fig. 3, neither a
ZFC-FC bifurcation nor a hump are visible due to negligible amount of
ferromagnetic domains. Also plotted (in blue) is the paramagnetic response for
free Gd spins $M=CH/T$ where $C=Ng^{2}S(S+1)\mu_{B}^{2}/(3k_{B})$ with $S=7/2$
and $N$ being the number of Gd ions per mole; it is clearly demonstrated that
paramagnetic Gd ions in the AFM region give the dominant contribution at lower
temperatures. As doping $x$ increases, ${\rm Gd}$ concentration decreases
resulting in smaller contribution to magnetization by the paramagnetic ${\rm
Gd}$ spins at lower temperatures (compare magnetization values for $x=0.6$
with those for $x=0.5$ at ${\rm T<T^{FC}_{min}}$ in Fig. 3).

It is also important to note that, since the charge-order energy scale (set by
$\beta E_{p}$ in Eq. (3)) is higher than both the ferromagnetic spin-order
energy scale (set by
$\left[t_{j,j+\delta}\cos(\theta_{j,j+\delta}/2)\right]^{2}/(2\gamma E_{p})$
in Eq. (3)) as well as the AFM energy scale $J_{ij}$ (mentioned in Eq. (4)),
the charge ordering occurs at a much higher temperature than the magnetic
ordering (as shown in Fig. 3).

Figure 5: (colour online) Temperature dependent magnetization curves of
Gd1-xCaxMnO3 \- 900 ($x$ = 0.4, 0.5, 0.6) nanoparticles under a field of 100
Oe. FC (ZFC) data is depicted in red (black). The continuous lines are meant
to be guides to the eye.

####  IV.1.2 Nano-form samples

Next, we will analyze the magnetization curves for manganites in nanoparticle
forms at the same dopings $x=0.4$, $0.5$ and $0.6$ as was done for the
manganites in bulk form. We did not observe any charge order for the nano
forms of GCMO reported in this paper; this is in contrast to the bulk case
where we witnessed CO (see Fig. 3). It is interesting to note that, for the
lower-doped cases among the nano forms considered, i.e., for $x=0.4$ and $0.5$
as shown in Fig. 5, we get negative magnetization at lower temperatures (i.e.,
$T\lesssim 20~{}{\rm K})$ for the FC case. In the nanoparticles, the unit
cells are less rigid at the surface of the nanoparticles compared to the
center. Consequently, it is natural that unit cells with smaller tolerance
factors (which lead to distortion effects such as more buckling of ${\rm Mn-O-
Mn}$ bonds) can be accommodated better at the surface than at the center.
Hence, since ${\rm Gd}$ ion has a smaller radius than the ${\rm Ca}$ ion, the
concentration at the surface (center) of the nanoparticle for the ${\rm Gd}$
ions is higher (lower) than the overall stoichiometric concentration given by
the formula Gd1-xCaxMnO3. Specifically, at $x=0.4$ ($0.5$), the outer shell
has less than $0.4$ ($0.5$) holes per unit cell. It is important to realize
that only when the effective concentration of ${\rm Gd}$ ions is more than
$0.5$ in a region can we get negative magnetization in that region; this is
because, then the average effective spin of ${\rm Gd}$ ions is $>7/4$ and the
average effective spins of ${\rm Mn}$ ions $<7/4$ and the spins of ${\rm Gd}$
ions align antiparallel to those of ${\rm Mn}$ ions which are in the direction
of the weak external magnetic field (H = 100 Oe). Furthermore, there is
greater charge delocalization at the surface because charge order is weaker
here. Consequently, the charge polarizes beyond the nearest neighbor unlike
the bulk situation modeled by Eq. (3); the manganite in the nanoparticle form
is more ferromagnetic than in the bulk form at the same effective
concentration because of stronger double exchange phenomenon. For these
reasons, compared to the bulk form, we have larger magnetism in nanoparticle
form at higher temperatures and also larger negative magnetization at low
temperatures. Accordingly, we see that the FC curves show large negative
magnetization at low temperatures with the uncoupled ${\rm Gd}$ spins in the
antiferromagnetic core making only a weak positive magnetization contribution
at low temperatures. Thus the observed magnetism is primarily from the outer
shell for the cases $x=0.4$ and $0.5$. Furthermore, for the above mentioned
reasons, the FC curve for the $x=0.6$ nano-system is similar to that of the
bulk ${\rm Gd_{0.5}Ca_{0.5}MnO_{3}}$ system with the peak magnetization being
much smaller than at $x=0.4$ and $0.5$.

As regards the ZFC curves at $x=0.4$, $0.5$ and $0.6$, since all the
ferromagnetic domains are not fully aligned with the external field, we
observe a weaker effect due to the outer shell. Unlike the FC curves, the ZFC
curves are non-negative due to the fact that ferromagnetism is weaker when
cooled in the absence of the external field; at lower temperatures, the
negative magnetization resulting from the ferromagnetic outer shell is weaker
than the positive magnetization resulting from the uncoupled ${\rm Gd}$ spins
in the antiferromagnetic core.

We now note that as particle size increases in various nano forms of GCMO, the
FC (ZFC) curves gradually tend toward the FC (ZFC) curves of the bulk system
as expected. In confirmation, as particle size increases in Fig. 6 for ${\rm
Gd_{0.5}Ca_{0.5}MnO_{3}}$, the size of the hump decreases and the negative
magnetization at lower temperatures reduces.

Figure 6: (colour online) Temperature dependent magnetization curves of
Gd0.5Ca0.5MnO3, under a field of 100 Oe, for different annealing temperatures
leading to different nanoparticle forms. The FC (ZFC) data is presented in red
(black).

We will now make the important observation that negative magnetization at
dopings $x=0.4$ and $0.5$ in nano-form GCMO is a metastable state because net
magnetization is opposite to the applied field. Also, we expect ZFC curves to
be further away from equilibrium than the FC curves because exposing the
samples to magnetic field (while cooling) in the FC case helps them tend
towards equilibrium. It is of interest to note that by-and-large the
magnetization curves for the FC case (shown in Figs. 3 and 5) and the remnant
magnetization ${\rm M_{R}}$ curves (depicted in Fig. 7) are similar at higher
temperatures (i.e., $T>20$ K). Thus, it is reasonable to expect that the FC
curves and the remnant magnetization curves (obtained under very different
conditions) are representative of the equilibrium physics at these higher
temperatures (i.e., $T>20$ K).

Figure 7: (colour online) Temperature dependence of remnant magnetization
${\rm M_{R}}$ and coercive field ${\rm H_{C}}$ obtained from magnetic
hysterisis loops.

###  IV.2 Remnant magnetization analysis at low temperatures

At lower temperatures (i.e., $T\lesssim 20~{}{\rm K})$, we will analyze the
remnant magnetization curves for the nanoparticle forms and the bulk forms
separately. The expectation at equilibrium is that the ${\rm Gd}$ spins in the
ferromagnetic regions will align with the external field whereas the ${\rm
Mn}$ spins in the same regions will be antiparallel due to their
antiferromagnetic coupling to the ${\rm Gd}$ spins. At 5 K, although the
nanoparticle forms in Fig. 7 display positive magnetization, they show much
larger magnetization than when ${\rm Gd}$ spins are parallel to the external
field and the ${\rm Mn}$ spins are antiparallel, i.e., much larger
magnetization than obtained by reversing the sign of the magnetization in the
nano-form curves for $x=0.4$ and $0.5$ in Fig. 5 at 5 K. Now, it is to be
noted that during the isothermal magnetization measurements, due to exposure
to high magnetic fields, the ${\rm Gd}$ spins in the ferromagnetic regions
will be aligned with the external field. Consequently, we infer that the ${\rm
Mn}$ spins are not aligned antiparallel to the external magnetic field. Hence,
the remnant magnetization curves also do not correspond to equilibrium.

As regards the bulk forms at $T\lesssim 20~{}{\rm K}$, for $x=0.5$, as
explained earlier, the magnetization due to ${\rm Gd}$ and ${\rm Mn}$ spins
cancel each other in the ferromagnetic regions at low temperatures; the only
contribution to magnetization is from the effectively uncoupled ${\rm Gd}$
spins in the antiferromagnetic region. Thus, the equilibrium value of
magnetization for $x=0.5$ is expected to be similar to the FC value in Fig. 3
which is slightly higher than the remnant magnetization in Fig. 7.

Next, the bulk form at $x=0.4$, in a unit cell, has an effective spin for the
${\rm Gd}$ ions $\big{(}=\frac{7(1-x)}{2}=2.1\big{)}$ which is larger than the
effective spin for the ${\rm Mn}$ ions
$\big{(}=2(1-x)+\frac{3x}{2}=1.8\big{)}$. Furthermore, the magnetization due
to ${\rm Gd}$ ions in the ferromagnetic (antiferromagnetic) regions ${\rm
M_{Gd}^{F}}$ (${\rm M_{Gd}^{AF}}$) and the magnetization due to the ${\rm Mn}$
ions in the ferromagnetic regions ${\rm M_{Mn}^{F}}$, together contribute as
$-{\rm M_{Gd}^{F}}+{\rm M_{Mn}^{F}}+{\rm M_{Gd}^{AF}}$ at 5 K to the total
magnetization ${\rm M_{tot}}$ in Fig. 3. Then, based on the FC curve for
$x=0.4$ in Fig. 3, we see that at 5 K ${\rm M_{tot}}\sim 20$ emu/mole, ${\rm
M_{Mn}^{F}}\sim 50$ emu/mole (based on FC curve value at 50 K), ${\rm
M_{Gd}^{F}}\sim 50\times 2.1/1.8\sim 60$ emu/mole; consequently, at 5 K ${\rm
M_{Gd}^{AF}}={\rm M_{tot}}-{\rm M_{Mn}^{F}}+{\rm M_{Gd}^{F}}\sim 30$ emu/mole.
For the bulk-form case of $x=0.4$ in Fig. 7, where the remnant magnetization
value is $\sim 100$ emu/mole at 5 K, it appears that the ${\rm Gd}$ spins in
the ferromagnetic and antiferromagnetic regions, when aligned parallel to the
external magnetic field, contribute as ${\rm M_{Gd}^{F}}+{\rm M_{Gd}^{AF}}\sim
90$ emu/mole. This implies that the contribution of the ${\rm Mn}$ spins is
$\sim 10$ emu/mole which is certainly not an equilibrium situation. At
equilibrium, the magnetization contribution of the ${\rm Mn}$ spins should be
negative (i.e., $\sim-50$ emu/mole).

It should also be noted that, as expected, the variation of the coercive field
as a function of the temperature in Fig. 7 follows a trend that is similar to
that of the remnant magnetization. Various details of the hysterisis loops are
given in Appendix A.

Figure 8: (colour online) Effect of different applied fields on the
temperature dependent magnetization curves of Gd0.5Ca0.5MnO3 \- Bulk and
Gd0.5Ca0.5MnO3 \- 1000 systems.

###  IV.3 FC and ZFC cases for stronger external fields

Lastly, we study the effect of stronger magnetic fields (i.e., fields stronger
than 100 Oe) on the temperature dependence of magnetization in both bulk and
in nanoparticle forms. In Fig. 8, we note that, as expected, at higher
magnetic fields both the FC and ZFC curves for bulk and nano forms of GCMO
samples increase in magnetization and the curves shift upwards. We also see
that, as applied magnetic field becomes stronger, in the lower-doped among the
nano compounds considered (such as ${\rm Gd_{0.5}Ca_{0.5}MnO_{3}}$), the
system tends to attain equilibrium state (with the Gd spins in the
ferromagnetic regions tending to align with the external field) thereby
becoming more prone to positive magnetization at low temperatures.
Additionally, as the magnetic field increases, the degree of ZFC-FC
bifurcation decreases; at sufficiently high fields, the FC and ZFC curves
merge. The effect of increasing magnetic fields on the magnetization for bulk
form of GCMO at different dopings is shown in Appendix B. Furthermore, as the
particle size increases in the lower-doped ones among the GCMO systems
considered (such as ${\rm Gd_{0.5}Ca_{0.5}MnO_{3}}$), there is a decrease in
the magnetic field needed to produce positive magnetization and make the
systems attain equilibrium at low temperatures, i.e., at $T\sim 5~{}{\rm K}$
(see Fig. 8 and Fig. 12 in Appendix C).

##  V Conclusion and future direction

In this work, by extending the framework developed earlier for ferromagnetic
insulators sp  to ferrimagnetic insulators, we explain negative magnetization.
Our picture of negative magnetization, invoking surface effects of
nanoparticles, is comprehensive and is valid not only at the doping levels
considered in this work (i.e., $x\geq 0.4$) but also at dopings lower than
those considered. We also provide an understanding of the size effect of the
particles on negative magnetization in GCMO of various forms (i.e., bulk,
thin-film and nano forms).

In future, experiments to probe the local concentration of Gd ions are needed
to further validate our picture that, near the surface of nanoparticles, the
concentration of Gd ions is higher than that indicated by the chemical fomula
Gd1-xCaxMnO3. Making thin films and using cross-sectional transmission
electron microscopy, to measure the surface Gd-ion concentration as a function
of depth, is likely to clarify the situation.

Figure 9: (colour online) Temperature dependent magnetization curves of
Dy0.6Ca0.4MnO3 for different particle sizes and annealing temperatures under a
field of 100 Oe.

It is also important to test the universal nature of our above-mentioned
picture of negative magnetization phenomenon when other high-magnetic-moment,
rare-earth ions are used in place of Gd ions in perovskite manganites. To this
end Dy0.6Ca0.4MnO3 has been studied. This compound has been selected because,
in the bulk form, it shows a slight ZFC-FC M(T) crossing but no negative
magnetization in the FC M(T) curve whereas negative magnetization is observed
at lower concentrations (i.e., $x$ = 0.2, 0.25, and 0.3) 5 .

Bulk form and variously-sized nanoparticle forms of Dy0.6Ca0.4MnO3 have been
prepared and magnetization studies have been carried out. Dy0.6Ca0.4MnO3 \-
bulk and Dy0.6Ca0.4MnO3 \- 900 nanoparticle form show small bifurcation with
small hump indicating ferromagnetic ordering. On further decreasing the
particle size, Dy0.6Ca0.4MnO3 \- 700 nanoparticle form showed ZFC-FC crossing
and negative magnetization in the FC curve (see Fig. 9). Thus, the emergence
of negative magnetization in the nano form of $R_{1-x}D_{x}$MnO3 perovskite
manganites (with $R^{3+}$ being a high-magnetic-moment rare-earth ion) by
controlling the system size seems to be a general phenomenon in manganites
which needs extensive investigation.

We would like to point out that, although Sm1-xSrxMnO3 (for $x\leq 0.05$)
shows negative magnetiztion ssmo , it belongs to a different type of systems
where a disordered paramagnetic sublattice is coupled with an ordered canted-
antiferromagnetic sublattice via negative-exchange interaction. We aim to
study such systems in future.

In future, we also would like to examine the deeper connection between
multiferroicity in $R{\rm MnO_{3}}$ compounds ($R{\rm=Gd,~{}Tb,~{}and~{}Dy}$)
kimura2 ; tokura1 ; kimura3 ; cheong2 ; arima  and negative magnetization that
results when $R{\rm MnO_{3}}$ systems are doped by divalent alkaline elements
Sr, Ca, etc.

Lastly, it would be useful to develop materials with temperature-induced
magnetization reversal in the vicinity of room temperature as they have
potential for application in magnetic memories such as thermally assisted
magnetic random access memory.

##  VI Acknowledgment

One of the authors (S.Y.) would like to acknowledge useful discussions with K.
Pradhan, I. Das, and G. Baskaran. Papri Dasgupta is thankful to the Department
of Science and Technology (DST), India for financially supporting this study
through Women Scientist Project (Ref. No. SR/WOS-A/PM-24/2017).

##  Appendix A Hysterisis loops

The hysteresis curves, depicted in Fig. 10, reveal the expected features that
as temperature decreases magnetization increases both in bulk and nano
samples; at lower temperatures, the curves tend towards saturation faster. The
unusual and interesting trends of the remnant magnetizations (obtained from
the hysteresis curves at various temperatures, dopings, and sample sizes) have
been discussed in the main text.

Figure 10: (colour online) Magnetization as a function of field at different
temperatures in bulk and nano-form samples of Gd1-xCaxMnO3 at dopings $x=0.4$
and $0.5$. Inset shows enlarged version near the origin.

##  Appendix B Bulk form magnetization at stronger magnetic fields and various
dopings

In Fig. 11, we depict that, as expected, the magnetization increases at higher
magnetic fields for both the FC and the ZFC cases in bulk forms of GCMO
samples; the curves shift upwards. Furthermore, the ZFC-FC bifurcation reduces
with increasing fields.

Figure 11: (colour online) Temperature dependent magnetization curves under
different applied magnetic fields in bulk Gd1-xCaxMnO3 (at $x=0.4$ and 0.5).
At any field, the FC curves have a higher peak.

##  Appendix C Transition from negative to positive magnetization due to
increasing magnetic field in nano-form GCMO

In Fig. 12 we demonstrate that, as the particle size increases in nano-form
Gd0.5Ca0.5MnO3, a smaller magnetic field is needed to produce a positive
magnetization at lower temperatures.

Figure 12: (colour online) Temperature variation of field-cooled magnetization
curves M(T) under different applied magnetic fields in Gd0.5Ca0.5MnO3 \- 800,
Gd0.5Ca0.5MnO3 \- 900, Gd0.5Ca0.5MnO3 \- 1000 and Gd0.5Ca0.5MnO3 \- 1100
nanoparticle forms.

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