---
abstract: 'We report detailed magnetization measurements on the perovskite oxide NdNiO$_3$. This system has a first order metal-insulator (M-I) transition at about 200 K which is associated with charge ordering. There is also a concurrent paramagnetic to antiferromagnetic spin ordering transition in the system. We show that the antiferromagnetic state of the nickel sublattice is spin canted. We also show that the concurrency of the charge ordering and spin ordering transitions is seen only while warming up the system from low temperature. The transitions are not concurrent while cooling the system through the M-I transition temperature. This is explained based on the fact that the charge ordering transition is first order while the spin ordering transition is continuous. In the magnetically ordered state the system exhibits ZFC-FC irreversibilities, as well as history-dependent magnetization and aging. Our analysis rules out the possibility of spin-glass or superparamagnetism and suggests that the irreversibilities originate from magnetocrystalline anisotropy and domain wall pinning.'
author:
- Devendra Kumar
- 'K. P. Rajeev'
- 'J. A. Alonso'
- 'M. J. Martínez-Lope'
title: 'Spin canted magnetism, decoupling of charge and spin ordering in NdNiO$_3$'
---

Introduction
============

The rare earth nickelates (RNiO$_3$, R $\neq $ La) have been under active investigation for the past two decades because of the interesting electronic and magnetic properties exhibited by these systems.@Medarde [@Catalan] These oxides undergo a bandwidth controlled metal-insulator (MI) transition on changing the temperature, chemical or hydrostatic pressure.@Torrance [@Obradors; @Canfield; @Zhou; @Ashutosh] In the metallic state the structure of these nickelates is that of an orthorhombic distorted perovskite with space group ${Pbnm}$.@P.Lcorre The metal to insulator transition occurs with a structural transition which consists of an increase in the unit cell volume, a decrease in Ni-O-Ni bond angle and a symmetry lowering from orthorhombic ${Pbnm}$ to monoclinic $P2_{1}/n$. The symmetry lowering is understood in terms of charge ordering with a charge disproportionation 2Ni$^{3+}\to$ Ni$^{3+\delta}+$ Ni$^{3-\delta}$ with $\delta \approx
0.2-0.3$.@Alonso_1 [@Alonso_2; @Alonso_3; @Alonso_4; @Staub; @Medarde1] In the early reports, the M-I transition of these compounds was attributed to the opening of an Ni-O charge transfer gap created by band narrowing.@Torrance But the occurrence of charge ordering at the M-I transition and some recent theoretical calculations suggest that the M-I transition owes its origin to the opening of a gap between the spin up $e_g$ band of Ni$^{3-\delta}$ and the hardly spin polarized $e_g$ band of Ni$^{3+\delta}$.@Mazin In these compounds the higher temperature phase is metallic and the lower temperature phase is insulating. The M-I transition is of first order and is associated with a large thermal hysteresis and time dependent effects in transport properties such as resistivity and thermopower.@Granados [@Devendra; @Devendra1; @Devendra2] During the cooling process, in the temperature window where hysteresis is seen, these compounds phase separate into insulating and supercooled metallic regions. The supercooled regions are metastable and they switch over to the insulating state stochastically giving rise to time dependence and hysteresis in transport properties.@Devendra [@Devendra1; @Devendra2] The nickelates also undergo a temperature driven continuous magnetic transition, which is relatively less studied, because the higher magnetic moment of rare earth ion (e.g. Nd$^{3+}$ moment $\approx 3.6\mu_B$) makes it difficult to get any information about the magnetic ordering of the Ni sublattice (Ni$^{3+}$ moment $\approx 1 \mu_B$) through magnetization measurements.@Blasco [@Perez; @Medarde] Muon spin rotation experiments of Torrance et al. show that these compounds undergo a magnetic ordering from paramagnetic to an antiferromagnetic state on lowering the temperature.@Torrance The magnetic ordering temperature ($T_{\textnormal{N}}$) coincides with the M-I transition temperature ($T_\textnormal{MI}$) for PrNiO$_3$ and NdNiO$_3$, while it is lower than $T_\textnormal{MI}$ for all the other nickelates.

Neutron diffraction experiments show that, below $T_\textnormal{N}$, the magnetic arrangement of Ni moments is characterized by the propagation vector (1/2,0,1/2) which suggests three possible magnetic structures, of which, two are collinear and one is non-collinear.@Garc [@Garc1; @Alonso_1; @Casais] The collinear magnetic structure consists of up-up down-down stacking of Ni magnetic moments, where each Ni moment is antiferromagnetically coupled to three of its nearest neighbors and ferromagnetically to the remaining three nearest neighbours. This magnetic structure implies that the orbital degeneracy of Ni$^{3+}$ $e_g^1$ electrons should be lifted by an orbital ordering, a prediction which has not gathered any experimental support so far.@Garc1 Soft X-ray resonant scattering experiments at the Ni $L_{ 2,3}$ edges show that the (1/2,0,1/2) reflections are purely of magnetic origin with no orbital contribution whatsoever thus more or less ruling out collinear magnetic order in the system.@Scagnoli In fact, the orbital degeneracy of the Ni$^{3+}$ $e_g^1$ electron is found to be lifted by a process of charge-ordering@Mazin and this supports the existence of a non-collinear magnetic structure which do not require any orbital ordering. The ambiguity of the magnetic structure of the nickelates is not yet completely resolved. However, the low temperature specific heat data and the resonant soft X-ray diffraction data of induced Nd magnetic moment in NdNiO$_3$ indicates that most likely the ordering of Ni moments in NdNiO$_3$ is non-collinear.@Bartolome [@Scagnoli1]

In this work, for the first time, we report the magnetization of the Ni *sublattice*, which we extracted after carefully subtracting the contribution of the Nd moments from the total magnetization. The magnetization of the Ni sublattice shows weak ferromagnetism which indicates that the magnetic arrangement of the Ni moments is perhaps canted. The existence of weak ferromagnetism cannot be understood in terms of the magnetic structures referred to in the previous paragraph, even the noncollinear ones. This suggests that these magnetic structures do not represent the true picture, and the actual magnetic arrangement of Ni moments could be quite different from what has been thought of so far. Further, we found that the supercooled metallic phases are magnetically ordered which shows that the transition from paramagnetic to antiferromagnetic state is complete at 200 K. This shows that the connection between the magnetic transition and the metal-insulator transition is rather week and they do decouple if the system system is supercooled. Also the magnetization of the Ni sublattice shows features such as FC-ZFC irreversibility which is indicative of the presence of frustration in the weak ferromagnetic state.

Experimental Details
====================

High quality polycrystalline NdNiO$_3$ pellets were prepared by a liquid mixture technique described elsewhere.@Massa All the magnetic measurements were performed in a SQUID magnetometer (Quantum Design, MPMS XL). To get the magnetization of Ni sublattice we subtracted the contribution of Nd moments from that of NdNiO$_3$. The Nd moment was estimated from magnetization data of NdGaO$_3$ which has a crystal structure similar to that of NdNiO$_3$. Since gallium and oxygen has zero magnetic moment, the magnetization of NdGaO$_3$ has contributions only of the Nd moments sitting at the A sites of the perovskite structure. On subtraction of NdGaO$_3$ magnetization (per mole) from that of NdNiO$_3$, we get the magnetization of the Ni sublattice. This subtraction is valid down to 50 K. Below this temperature the Ni sublattice of NdNiO$_3$ induces a partial Nd ordering and so the data of NdGaO$_3$, which has no such ordering, does not give an accurate estimate of the contribution of Nd moment in NdNiO$_3$.@Garc1

The field dependent resistivity measurements were performed in a home made cryostat placed inside an electromagnet.

![(Color Online) The temperature variation of the magnetization of NdNiO$_3$ in FCC (circles), FC (triangles), and in ZFC (squares) protocol at 500 Oe. The triangles are behind the circles and hence their view is somewhat obscured. The stars show the magnetization of NdGaO$_3$ at the same field. For NdGaO$_3$ the FCC, FC and ZFC magnetizations coincide.<span data-label="fig: M raw"></span>](fig94.eps){width="0.8\columnwidth"}

\[!t\]

![(Color Online) The dc magnetic susceptibility of the Ni sublattice versus temperature for FC and ZFC protocols at various fields. Inset of (a) shows how $T_\textnormal{IRR}$ and $T_\textnormal{P}$ depend on applied field. In inset of (b) we show the temperature dependence of resistivity at zero field and 1000 Oe during cooling as well as warming.<span data-label="fig: M vs T"></span>](fig18.eps){width="0.8\columnwidth"}

Results and Discussion
======================

### Magnetization measurements

Figure \[fig: M raw\] shows the magnetization of NdNiO$_3$ and NdGaO$_3$ in FC, ZFC and FCC protocols at 500 Oe. In FC protocol we cool the sample in the presence of a specified field and then record the magnetization in the subsequent heating run keeping the field unchanged. In ZFC protocol we cool the sample in zero field to the lowest temperature and then apply the specified field and record the magnetization while warming up. In FCC protocol the magnetization is recorded while cooling in the specified field. The magnetization plots of NdNiO$_3$ show a shoulder below 200 K which is suggestive of the ordering of Ni moments. We can see that below 200 K, the magnetization of NdNiO$_3$ depends on the experimental protocol. The FCC magnetization is slightly higher than the FC magnetization while ZFC magnetization is lower than both FCC and FC magnetizations. Above 200 K, the FCC, FC, and ZFC magnetization curves overlap and are indistinguishable. The existence of thermal and magnetic history dependence in magnetization suggests that the system is not in thermodynamic equilibrium below 200 K. In contrast, for the reference sample NdGaO$_3$, the magnetization values in FCC, FC, and ZFC protocol coincide and follow the Curie law.

![(Color Online) M-H curves for the Ni sublattice at 150 K and 60 K. The inset shows the expanded view of low field data.<span data-label="fig: M vs H"></span>](fig71.eps){width="0.8\columnwidth"}

<span>|c|c|c|c|c|c|</span>   Field (Oe) & C & $\theta$ & $\chi_0$ & $\chi^{2}/DOF$ & $R^{2}$  2000(ZFC) & 0.017(3) & 163(6) & 0.00111(2) & 0.3074 & 0.99953  1000(FC) & 0.043(4) & 125(6) & 0.00095(2) & 1.314 & 0.99916  1000(ZFC) & 0.043(4) & 126(6) & 0.00095(2) & 1.271 & 0.99920  500(FC) & 0.04(2) & 132(21) & 0.0011(1) & 0.6957 & 0.99871 500(ZFC) & 0.04(1) & 136(19) & 0.0011(1) & 0.6340 & 0.99874 200(FC) & 0.03(1) & 148(12) & 0.0012(1) & 0.4894 & 0.99905 200(ZFC) & 0.03(1) & 150(14) & 0.0012(1) & 0.6333 & 0.99863

### Magnetic ordering of the Ni sublattice

Figure \[fig: M vs T\] shows the temperature dependence of ZFC and FC dc magnetic susceptibility of Ni sublattice between 100 Oe to 2000 Oe. Above 220 K the data fit well to the Curie Weiss equation $$\chi = C/
(T-\theta) + \chi_{0} \label{eq:Curie Weiss}$$ where $C$ and $\theta$ are Curie and Weiss constants respectively, and $\chi_{0}$ is a constant arising from Van Vleck and Pauli paramagnetism and Landau and core diamagnetism. The Pauli and Landau contributions arise from itinerant electrons. The parameters obtained from the fitting of equation \[eq:Curie Weiss\] to the susceptibility data in the temperature range of 220-300 K is shown in table \[tab:fitting-parameters\]. It is to be noted that we get a positive Weiss constant $\theta$ which is indicative of a ferromagnetic interaction in the magnetically ordered state. This is surprising considering the fact that neutron diffraction measurements show that the system has antiferromagnetic ordering below $T_\textnormal{MI}$. In consonace with the above observation we note that below 195 K, in FC measurements, the magnetic susceptibility increases on decreasing the temperature as would be expected in the case of ferrimagnets or canted antiferromagnets.

In figure \[fig: M vs H\] and its inset we show the magnetization versus field for the Ni sublattice at 150 K and 60 K. The M-H curves show a small hysteresis at small fields and at higher fields behave as that of a typical antiferromagnet, with $M$ varying linearly with $H$, which leads to the conclusion that this system is a spin-canted antiferromagnet.@chikazumi

Referring to the inset of figure \[fig: M vs H\], we see that, the coercivity ($H_\textnormal{C}$) is temperature dependent and it increases on lowering the temperature below $T_\textnormal{N}$. Since coercivity is related to magnetic anisotropy, this suggests that the magnetic anisotropy increases on decreasing the temperature.

![(Color Online) The temperature variation of the difference in magnetization, $M_\textnormal{FCC} - M_\textnormal{FC}$, of NdNiO$_3$ between cooling and heating runs at 100 Oe, 200 Oe, 500 Oe and 2000 Oe (solid symbols). The open circles show the difference in the metallic volume fraction, $V_\textnormal{C}-V_\textnormal{H}$, between cooling and heating runs.<span data-label="fig: M diff"></span>](fig85.eps){width="1.0\columnwidth"}

### Magnetic state of supercooled phase

The transport properties of NdNiO$_3$ show thermal hysteresis which is attributed to the presence of supercooled metallic phases below the transition temperature.@Devendra [@Devendra1; @Devendra2] The magnetic state of the supercooled metallic regions is not clear in the sense that we do not know whether it is paramagnetic or antiferromagnetic. In order to throw some light on this issue we measured the thermal hysteresis of magnetization. In figure \[fig: M diff\] we show the difference in cooling and heating cycle magnetization, $M_\textnormal{FCC} - M_\textnormal{FC}$, of NdNiO$_3$ at a few field values in the range 100 Oe to 2000 Oe. The data show that between 200 K and 120 K, the magnetization of the cooling cycle is higher than that of the heating cycle. The difference in the magnetization is maximum around 170 K. Figure \[fig: M diff\] also shows the difference in the metallic volume fractions between the cooling and heating runs $V_\textnormal{C}-V_\textnormal{H}$, taken from reference . The difference in the magnetizations and the difference in the metallic volume fractions have remarkably similar temperature dependence which suggests that they originate from the same physical mechanism. In a cooling run, below $T_\textnormal{MI}$, the system contains supercooled metallic and insulating regions, while in a heating run, it is mostly insulating.@Devendra [@Devendra1; @Devendra2] Therefore $V_\textnormal{C}-V_\textnormal{H}$ represents the volume fraction of supercooled metallic regions. So the correlation between the thermal hysteresis in magnetization and the supercooled metallic volume fraction indicates that the supercooled metallic regions have a higher magnetic moment compared to the insulating regions.

The Ni moments are paramagnetic in the normal metallic state ($T>T_\textnormal{MI}$) while they show a spin-canted antiferromagnetic ordering in the insulating state. Also, the spin-canted insulating state has a higher susceptibility than the paramagnetic metallic state (Figure \[fig: M vs T\]). This suggests that if the supercooled metallic regions were paramagnetic, as above $T_\textnormal{MI}$, then the magnetization of NdNiO$_3$ in a cooling run, where below $T_\textnormal{MI}$ the system consists of supercooled metallic and insulating regions, should be lower than that in a heating run where the system is expected to be almost fully insulating. But the experimental results discussed in the previous paragraph contradict this which indicates that the supercooled metallic regions are not paramagnetic. To make things more concrete, we can compare the observed difference in the magnetization of cooling and heating runs to the expected value of the difference if the supercooled regions were paramagnetic. In the cooling run, at 170 K, the volume fraction of the supercooled metallic regions is around 0.9 from figure \[fig: M diff\]. The dc magnetic susceptibility of the paramagnetic metallic phase at 2000 Oe (Figure \[fig: M vs T\]), extrapolated down to 170 K, is about 20% smaller than that of the insulating phase which suggests that if the thermal hysteresis in the magnetization is because of paramagnetic ordering of supercooled metallic regions, then, according to our estimate, the difference in the magnetization of the cooling and heating runs should be around $-0.9$ emu/mole. But the observed difference in the magnetization is $+0.4$ emu/mole which has the wrong sign and is smaller in magnitude than the expected value. This observation strongly suggests that the supercooled metallic regions are antiferromagnetic with canted spins just like the insulating state. The small positive difference in magnetization between cooling and heating is proportional to the volume fraction of supercooled metallic regions and hence we conclude that this difference in susceptibility is temperature independent. This suggests that the observed difference in cooling and heating cycle magnetization is coming from itinerant electrons in the metallic state through Pauli paramagnetic and Landau diamagnetic contributions.@endnote1 Thus we see that the metallic state has a paramagnetic ordering above $T_\textnormal{MI}$ and on cooling below $T_\textnormal{MI}$, while a fraction of the high temperature metallic phase exists in its supercooled state, the magnetic ordering of the whole sample switches to an antiferromagnetic state at $T_\textnormal{N}$.

From the above discussion it can be concluded that charge ordering which results in a metal insulator transition and magnetic ordering are not strongly coupled and these transitions occur independently of each other. This result is also consistent with the fact that the magnetic ordering transition is a continuous phase transition and hence there can be no metastable states associated with it. Incidentally, we note that, except in PrNiO$_3$ and NdNiO$_3$ of the nickelate series, the two transitions occur at different temperatures which supports the conclusion that the transitions are only weekly coupled.

![(Color Online) Memory experiment in the FC protocol with intermediate stops of one hour at 175, 150, 125 and 110 K. The field is switched off during each stop. The data close to 175 K is shown here. The black squares show the FC reference which is the magnetization in FCC protocol(after removing the contribution of thermal hysteresis).<span data-label="fig: FC memory"></span>](fig111.eps){width="0.9\columnwidth"}

![(Color Online) The time decay of thermoremanent magnetization of NdNiO$_3$ at 80 K (red circles). The blue squares show the decay of thermoremanent magnetization with a one hour wait time. The inset shows the response of the system to a 100 Oe applied field in ZFC protocol at 80 K.<span data-label="fig: TRM"></span>](fig44.eps){width="0.8\columnwidth"}

### The FC-ZFC irreversibility

The FC and ZFC magnetic susceptibilities show a history dependence with a bifurcation between the two curves at a temperature known as the temperature of irreversibility ($T_\textnormal{IRR}$). See figure \[fig: M vs T\]. The temperature of irreversibility depends on the magnetic field and it decreases on increasing the magnetic field. For fields greater than 2 kOe the FC and ZFC curves superpose. Behavior such as this where the magnetic susceptibility depends on measurement history has been observed in non-equilibrium systems such as spin-glasses,@Maydosh [@Tiwari; @Vijay] superparamagnets,@Knobel cluster-glasses,@Deac [@Huang] supercooled systems,@Chaddah [@Chaddah1] and also in ferromagnets and ferrimagnets.@Anil [@Anil1; @Anil2; @Song; @Roshkoa] The ZFC data show a peak, and the peak broadens and shifts to low temperatures on increasing the magnetic field. We analysed the nature of this peak and found that the peak temperature ($T_\textnormal{P}$) as a function of field ($H$) does not behave as in the case of spin-glasses, cluster-glasses,@Almeida or superparamagnets@Zheng [@Vijay1] which indicates that the system is neither a spin-glass nor a superparamagnet. We also rule out supercooling as a possible reason for the FC-ZFC irreversibility by the following argument. The resistivity measurements show a thermal history dependence which is attributed to the presence of supercooled metallic regions below $T_\textnormal{MI}$. We did not observe any significant magnetic field or magnetic history dependence in transport properties which suggests that the volume fraction of supercooled metallic regions is not altered by the application of a magnetic field. See Inset of figure \[fig: M vs T\](b). The lack of dependence of resistivity on applied magnetic field has also been reported earlier by Mallik et al.@Mallik From these results, we infer that the magnetic history dependence of the dc magnetic susceptibility (see figure \[fig: M vs T\](a) and (b)) cannot be originating from the supercooled metallic phases. So far our analysis has shown that the magnetic hysteresis does not arise from spin-glass, cluster-glass, superparamagnetism or supercooling. This leaves us with the only possibility that the magnetic hysteresis in this system is arising from magnetic anisotropy of the spin canted magnetic domains.

\[!t\]

![image](energy2.eps){width="2.0\columnwidth"}

To be doubly sure that the history dependent FC and ZFC susceptibility of the Ni sublattice has nothing to do with super-paramagnetism or spin-glassiness, we performed FC, ZFC memory and aging experiments. Since the Nd moments are paramagnetic, they would not have any role in the memory and aging of NdNiO$_3$. Thus if any such effect is seen in this system it would have to be attributed to the Ni sublattice. The FC memory experiments were performed with intermediate stops of one hour at 175, 150, 125, and 100 K. In these experiments the system is cooled in a 100 Oe field from 220 K to 80 K and then heated back to 180 K to remove the influence of supercooled metastable regions on dynamic behavior. Subsequently the system is cooled from 180 K to 80 K with intermediate stops of 1 hour at 175, 150, 125, and 100 K. The field was switched off during the intermediate stops. The magnetization is recorded while cooling and then during the subsequent heating. The FC memory data at 175 K is shown in figure \[fig: FC memory\]. We can see that immediately after an intermediate stop the magnetization does not go back to its pre-stop value after switching on the field. In the subsequent heating run, we did not find any memory of the intermediate stops and this rules out the possibility of superparamagnetism or spin-glass state in the system.@Vijay We also carried out ZFC memory experiments on the system at 170 K and the result was negative. This confirms the conclusions we arrived at from the FC memory experiments and once again rules out a spin-glass state.@Vijay

In figure \[fig: TRM\] we show the results of the FC ageing experiment. In this experiment one essentially measures the time decay of thermoremanent magnetization along with wait time dependence. To begin with we cool the system from 250 K to 80 K in the presence of 100 Oe field, wait for the duration $t_\textnormal{w}$ at 80 K with the field on, and then switch off the field and record the magnetization as a function of time. It is clear from the figure that the system does not show any noticeable wait time dependence in FC ageing and this yet again rules out the possibility of the system being a spin glass or a superparamagnet.@Vijay

The irreversibility of the FC and ZFC magnetic susceptibility in a system which is neither a spin-glass nor superparamagnetic can be understood in terms of a competition between the magnetocrystalline anisotropy and domain wall pinning on the one hand and applied field and thermal energy on the other.@Anil [@Anil1; @Anil2; @Song; @Roshkoa] Below the temperature of magnetic ordering, a magnetically ordered material consists of uniformly magnetized regions which are known as magnetic domains. At any temperature $T$ and applied field $H$, the free energy of the magnetic systems have a number of local minima which are determined by the arrangement of the domains inside the magnetic material. These local minima states are separated by energy barriers which arise due to magnetocrystalline anisotropy and domain wall pinning. When the thermal energy is greater than the energy barrier of the metastable state in which the system is trapped, the system can explore the neighboring states in search of the global minimum or the equilibrium state. The free energy configuration is a function of applied magnetic field $H$ and temperature $T$ and on changing $H$ or $T$ (which changes the magnetocrystalline anisotropy) the system evolves from one configuration to another.@Bertotti We shall make an attempt to understand our system on the basis of the Preisach model in which the free energy configuration is decomposed into an ensemble of bistable subsystems.@Bertotti A bistable subsystem consists of two metastable states separated by an energy barrier. The two states have moments oriented in opposite directions and are termed as $\pm \mu$ states. The free energy of these states in the absence of applied magnetic field is determined by the local interaction field ($H_\textnormal{S}$) and the the coercive field ($H_\textnormal{C}$). $H_\textnormal{S}$ is the net magnetic field produced at the location of the moment $\mu$ by the magnetic moments of all the neighboring domains. If $H_\textnormal{S}=0$ then $\mu H_\textnormal{C}$ represents the anisotropy energy barrier that has to be crossed to go from $+\mu$ to $-\mu$ state or vice versa. The barrier height seen from the $+\mu$ side is $\mu(H_\textnormal{C}+ H_\textnormal{S})$ while from the $-\mu$ side it is $\mu(H_\textnormal{C}- H_\textnormal{S})$. See figure \[fig: energy\] (a). The application of a magnetic field ($H$) changes the free energy of the metastable states which in turn affects the effective height of the energy barrier. We also note that a change in the temperature can also affect the free energy barrier through its effect on magnetocrystalline anisotropy. @Song [@Roshkoa; @Bertotti]

In the following paragraphs we discuss qualitatively the FC-ZFC irreversibility and the remanent magnetization using the standard Preisach model. Thereafter we apply it to understand the observed results of aging experiments.

In ZFC protocol when the system is cooled below $T_\textnormal{N}$ each subsystem will be in its lower energy state which is determined by $H_\textnormal{S}$ (Figure \[fig: energy\] (a)). On applying a magnetic field, depending on the direction and strength of the applied field, the low energy state of the subsystem may remain as the low energy state (Figure \[fig: energy\] (b)), or may become metastable or unstable (Figure \[fig: energy\] (c) and (d)). If $\vec{H} \parallel -\vec {H_\textnormal{S}}$, the subsystems for which $H$ is larger than $H_\textnormal{C}+H_\textnormal{S}$, will flip to their new low energy state (Figure \[fig: energy\] (d)). It is this flipping that gives rise to the initial value of the ZFC magnetization of the system. The subsystems for which $H$ is less than $(H_\textnormal{C} + H_\textnormal{S})$, are now in a metastable state (Figure \[fig: energy\] (c)). These subsystems will undergo a thermally activated transformation, which gives rise to a slowly rising time dependent ZFC magnetization even if the magnetic field is held fixed. On increasing the temperature, $H_\textnormal{C}$ decreases and because of this more number of subsystems will flip to their new low energy state and this increases the ZFC magnetization further. As one increases the temperature the ZFC magnetization curve will attain a peak when the most probable $H_\textnormal{C}$ value of the Barkhausen moment ($\mu$) becomes equal to the applied field $H$.

In the FC protocol the subsystems get trapped in their low energy states, as the sample is cooled through the magnetic ordering temperature in the presence of an applied field. At a constant field, a decrease in temperature increases the energy barrier (because of increase in $H_\textnormal{C}$), but this does not affect the relative positions of the $+\mu$ and $-\mu$ states. Thus in the FC protocol there is hardly any change of state of the bistable subsystems when cooling through $T_N$. The temperature dependence observed in the FC magnetization is because of temperature dependence of the Barkhausen moment $\mu(T)$. That is why the shape of an FC magnetization curve is nearly the same for all fields.

If we switch off the applied field in the FC protocol, the subsystems for which applied field $\vec{H}$ is opposite and greater in magnitude than $\vec{H_\textnormal{S}}$ will result in their low energy state becoming a high energy state and vice versa. This can be understood looking at figure \[fig: energy\] where the initial states shown in figures \[fig: energy\] (c) or (d) switch to the final state shown in figure \[fig: energy\] (a)) on removal of the applied field. Of these subsystems, those which have $H_\textnormal{S}\geq H_\textnormal{C}$, will become unstable on removing the field, and their change of state constitutes the initial loss of FC magnetization. The other subsystems (which have $H_\textnormal{S}<H_\textnormal{C}$) will become metastable and their thermally activated transformation from a metastable to a new lower energy state gives a further slow decay in FC magnetization.

At this point let us examine the effect of aging (wait time dependence) on the system. All the subsystems occupy their lower energy state on cooling through $T_\textnormal{N}$. Thus, after cooling, if we wait for a few hours before switching off (or on) the field, it will not affect the population of the $\pm \mu$ states and hence we would not get any effect of aging on magnetic relaxation.

Conclusion
==========

We performed detailed magnetization measurements on NdNiO$_3$ and extracted the magnetization of Ni sublattice after removing the contribution of the rare earth Nd ion. Our results indicate the presence of weak ferromagnetism in the antiferromagnetically ordered Ni sublattice. We argued that the weak ferromagnetism is due to canting of antiferromagnetic spins. Further we found that in contrast to the normal metallic state, the supercooled metallic regions are magnetically ordered. This shows that while the metal insulator transition occurs over a temperature range of 200 K to 110 K, the magnetic ordering is complete at 200 K. Below $T_\textnormal{N}$, the ZFC-FC magnetizations diverge exhibiting irreversibilities, that could remind one of a spin-glass state. However, our analysis shows that the system is neither a spin-glass nor a superparamagnet, and the irreversibilities arise from the temperature-dependent magnetocrystalline anisotropy and domain-wall pinning.

Acknowledgements
================

DK thanks the University Grants Commission of India for financial support. JAA and MJM-L acknowledge the Spanish Ministry of Education for funding the Project MAT2010-16404.

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