Random magnetic anisotropy driven transitions in layered perovskite LaSrCoO₄

Low-dimensional magnetic systems are susceptible to a variety of external perturbations, such as doping, magnetic field, and strain, etc. [1]. Their higher sensitivity to external stimuli compared to the 3D counterparts opens up room for new applications, $e.g.$, control of exchange bias in a single unit cell [2]. Moreover, in low-dimensional systems, control of spin moment can provide tremendous application opportunities for spintronics devices (layered cobaltates are potential candidates). LaCoO₃ (Co³⁺), an often studied 3D compound, exhibits peculiar properties such as a metal-insulator transition, high-spin (HS) state to low-spin (LS) state transitions, magneto-electronic phase separation and ferromagnetic (FM) correlations [3]. A fascinating cobaltate and its 2D analog, La₂CoO₄ (Co²⁺) is an antiferromagnetic insulator (AFI) [4], but not explored enough mainly due to its topotactic oxidation [5]. Its counterpart Sr₂CoO₄ (Co⁴⁺) has a similar structure and shows ferromagnetism and half-metallicity [6]. In LaSrCoO₄ (an intermediate composition of the above-mentioned end compounds), La/SrCoO₃ 3D blocks are separated by rock-salt La/SrO layers.

Depending upon how the energy-splitting $\Delta_{CF}$ between the $t_{2g}$ and $e_{g}$ orbitals, caused by the crystal-field, compares in magnitude with the exchange energy $J_{ex}$ associated with the Hund’s rule coupling, the Co³⁺ ion can exist in three different spin states [7]: the magnetically-active Co³⁺ high-spin (HS; $t_{2g}^{4}e_{g}^{2}$, S = 2) state when $\Delta_{CF}$ $<$ $J_{ex}$, intermediate-spin (IS; $t_{2g}^{5}e_{g}^{1}$, S = 1) state when $\Delta_{CF}$ $\sim$ $J_{ex}$ and non-magnetic low-spin (LS; $t_{2g}^{6}e_{g}^{0}$, S = 0) state when $\Delta_{CF}$ $\gg$ $J_{ex}$. In the LaSrCoO₄ (LSCO) compound, the spin-state of Co³⁺ ions has been a controversial issue. For instance, LSCO is reported [8, 9, 10, 11, 12] to have a mixture of LS and (thermally-activated) HS states of the Co³⁺ ions. Presence of homogeneous IS state of Co³⁺ ions has also been claimed [13] and subsequently challenged [14]. In our previous study [7], we probed the spin-states in La_2-xSr_xCoO₄ (0.5 $\leq$ x $\leq$ 1) compounds and found existence of the HS and LS states together at room temperature, discarding the presence of IS state. First-principles calculations  [9] support the view that mixed spin-states (HS and LS) of Co³⁺ ions inhabit the ground state of LaSrCoO₄.

Existence of mixed spin-states of the Co³⁺ ions in LSCO is expected to have an important bearing on the magnetic and transport properties, since the super-exchange interaction between the HS Co³⁺ ions mediated by the intervening LS Co³⁺ ion can induce long-range FM order, while the superexchange interaction between the adjacent HS Co³⁺ neighbors can give rise to antiferromagnetic (AFM) order. The competing FM and AFM exchange interactions, in turn, could result in a spin glass (SG) state. Conflicting reports  [15, 16, 17, 18, 19, 20] about the nature of magnetism in LSCO, however, render the existing magnetic data inconclusive, as elucidated below. While a paramagnetic (PM)-SG transition at $T_{SG}$ $\sim$ 20 K [15] ($T_{SG}$ $\sim$ 8 K [16]) is construed as a transition from the HS to IS state, no PM-SG transition is observed down to 2 K (the temperature up to which an insulating PM state persists [18]) and a broad hump in the thermomagnetic data, observed at $\simeq$ 250 K, is attributed to the ferromagnetic (La,Sr)CoO₃ impurity [18]. This insulating PM ground state is taken to reflect [18] the IS spin state of Co³⁺ ions. In sharp contrast, two magnetic transitions, PM-FM transition at T_c $\simeq$ 250 K [19, 20] and the SG-like transition at $T_{SG}$ $\sim$ 12 K [19] or at $T_{SG}$ $\sim$ 60 K [20], have been reported. Considering that the spin and valence states of Co are inextricably linked in cobaltates, presence of valence states (Co²⁺ and/or Co⁴⁺) of Co, arising from the varying degree of oxygen off-stoichiometry in the LSCO samples used in such investigations, could be at the root of these apparent discrepancies.

In this work, the intrinsic magnetic behavior near the PM - FM and SG transitions is unraveled by ensuring close to a pure Co³⁺ state in our LaSrCoO₄ sample. DC magnetization and frequency-dependent ac susceptibility (ACS) data, taken on the LaSrCoO₄ compound, unambiguously confirm the existence of two magnetic transitions: a thermodynamic (frequency-independent) PM - FM phase transition at T_c = 220.5 K and a frequency-dependent cluster spin glass (SG) transition, approaching the value T_g = 7.7 K in the zero-frequency limit. Both of these transitions are basically driven by random magnetic anisotropy (RMA). Increasing strength of the single-ion magnetocrystalline anisotropy (and hence RMA) with decreasing temperature is shown to be a manifestation of an increase in the number of low-spin (LS) Co³⁺ ions at the expense of that of high-spin (HS) Co³⁺ ions.

The details of sample synthesis are reported elsewhere [7]. Synchrotron X-ray diffraction (XRD) pattern was measured at room temperature using the BL-12 ADXRD beamline of RRCAT, India. To avoid possible fluorescence from Co, an x-ray photon energy of 7690 eV ($\lambda_{XRD}$ = 1.612  Å), which lies below the absorption edge, was used for XRD measurements. X-ray absorption near edge spectra (XANES) were recorded in the fluorescence mode at the same beamline. Energy resolution at the Co K-edge energy was $\sim$ 0.7 eV. ‘Zero-field’ (H = 0) and ‘in-field’ (H = 80 kOe) electrical resistivities as functions of temperature were measured by four-probe method, using the Oxford cryostat. Magnetization measurements were carried out utilizing the 7 Tesla Quantum design MPMS 3 magnetometer. M(H) hysteresis loops at fixed temperatures were recorded following the ‘field-cooled’ (FC) protocol. Magnetization was measured as a function of temperature, M(T), over the temperature range extending from 5 K to 300 K at fixed fields, employing the FC and ‘zero-field-cooled’ (ZFC) protocols. The virgin M(H) curves were recorded after demagnetizing the sample and the superconducting magnet by setting field to zero in the oscillatory mode. For constructing the Arrott plots from the virgin M(H) isotherms, the applied field has been corrected for the demagnetizing field to arrive at the internal effective magnetic field [21] H_eff, in accordance with the relation, H_eff = H_applied - (4$\pi$N) M, where N is the demagnetizing factor, which was determined from the low-field portions of the M(H) isotherms taken at T $<$ T_c. ZFC relaxation measurement has been performed by cooling the sample down to 150 K in zero field and after waiting for 100 s (t_w), a static magnetic field of 50 Oe was applied and the time evolution of magnetization at 150 K was measured upto $\sim$ 8000 s. The real ($\chi^{\prime}(\omega,T)$) and imaginary ($\chi^{\prime\prime}(\omega,T)$) components of ac magnetic susceptibility were measured in the temperature range of 2 - 300 K, at fixed frequencies (1.1 Hz - 941 Hz) of the ac driving field of rms amplitude, $h_{ac}$= 3 Oe employing the Quantum Design MPMS 3 system equipped with an ac susceptibility option.

Fig. 1 shows room temperature synchrotron XRD pattern of the LaSrCoO₄ (LSCO) compound together with the Rietveld fit. Absence of any extra Bragg peaks, attributable to the impurity phases such as La₂CoO₄ (Co²⁺) and Sr₂CoO₄ (Co⁴⁺), confirms the single-phase nature of the LaSrCoO₄ sample. Rietveld refinement yielded the lattice parameters, corresponding to the tetragonal space group $I$4/$mmm$, as $a$ = 3.8025(3) Å  and $c$ = 12.4985(2) Å . Inset (a) compares the Co K-edge XANES data of LaSrCoO₄ with those of the reference compounds CoO and Co₃O₄. Linear Co-valency versus Co K-edge photon energy plot in the inset (i) yields the value +2.98(2) for the Co valence and thereby confirms the Co-valency of +3 in the LSCO sample. Data points, depicted by circles in the linear plot shown in inset (i), correspond to the first derivative of the absorption edge.

Inset (b) of Fig. 1 clearly demonstrates semiconducting-like temperature variation of the ‘zero-field’ ($\rho$(H = 0)) and ‘in-field’ ($\rho$(H = 80 kOe)) resistivity in the LSCO sample. If, apart from Co³⁺ ions, Co²⁺ and Co⁴⁺ ions were also present in the sample, double exchange, involving charge hopping, would have resulted in a metallic state (characterized by low resistivity, typically $\sim$ 10 - 100 $\mu\Omega~{}cm$, and $\rho$ increasing with temperature) as opposed to the observed semiconducting-like state with very large resistivity decreasing from $10^{6}~{}\Omega~{}cm$ at 130 K to $10^{2}~{}\Omega~{}cm$ at 310 K. Consistent with the conclusion drawn from the Co K-edge XANES data, this finding completely rules out the presence of Co²⁺ and Co⁴⁺ ions in our LSCO sample. Furthermore, negligibly small magnetoresistance (MR), [($\rho$(T,H = 80 kOe) - $\rho$(T,H = 0)]/$\rho$(T,H = 0), observed in the present case, sharply contrasts the reasonably large negative MR, expected in double-exchange ferromagnets. The insets (ii) and (iii) of Fig. 1(b) bear out clearly that the processes such as the thermal activation of charge carriers across the band gap (described by expression $\rho$ $\sim$ $e^{E/k_{B}T}$) and Mott variable-range polaron hopping (represented by the expression $\rho$ $\sim$ $e^{(T_{0}/T)^{1/4}}$) [22, 23, 24] operate within the temperature ranges 225 K - 300 K (T $>$ T_c) and 130 K - 220 K (T $\leq$ T_c), respectively.

After ensuring that the sample is of very good quality (free from the impurity phases), in the following, we attempt to unravel the intrinsic magnetic behavior of the LaSrCoO₄ compound. Fig. 2(a) displays the ‘field-cooled’ (FC) and ‘zero-field-cooled’ (ZFC) thermomagnetic curves, M(T), at fixed magnetic fields in the range 10 Oe - 10 kOe. A clear bifurcation between FC and ZFC M(T) is observed at low fields. Such bifurcations are indicative of either a spin-glass (SG) phase [25] or cluster spin-glass (CG) phase or magnetic anisotropy energy barriers or super-paramagnetism (SPM)  [26]. A closer look at the low-temperature ZFC M(T) data shows a clear cusp (inset (b) of Fig. 2) at T_g $\sim$ 8 K. Similar feature was observed previously [17, 19, 16] and denoted as a freezing temperature. An interesting point to note is that on cooling below T_Irr (the temperature at which FC and ZFC M(T) bifurcate), M_FC increases, which is taken to be a characteristic feature of cluster spin glass behavior in various systems [27]. The increase in M_FC with decreasing temperature, even below T_g, points to the presence of a cluster spin glass (CSG) state [28] because M_FC normally exhibits a plateau for canonical spin glass [29]. It is noteworthy to mention that, for a typical re-entrant SG systems, the irreversibility occurs far below T_c while for CSG T_Irr $\leq$ T_c [30]. In this connection, it is also reported [3] that if the bifurcation temperature T_Irr $>>$ T_g, the compound exhibits cluster spin glass type behavior while, for canonical SG [31], the T_Irr coincides with the T_g. Thus, we conclude that the signatures discussed above basically reflect the CSG state in LaSrCoO₄ at low temperatures.

Next, we discuss the role of magnetic anisotropy as a possible source of bifurcation in M(T). Because of the prevalent elongated octahedra in the crystal structures similar to that of LaSrCoO₄, the pseudo orbital moment of Co³⁺ HS, $\tilde{L}$ = 1 [32], prefers to lie in-plane and forces the spin moment also to lie within the plane [33] due to the spin-orbit coupling. Neutron diffraction studies on iso-structural compounds LaSrFeO₄ [34] and half-doped La_1.5Sr_0.5CoO₄  [35] have clearly borne out that, in these systems, magnetic moments are confined to the $ab$ plane but can rotate within the plane. This is indicative of an XY-type anisotropy and hence $c$-axis may not be the easy axis of magnetization. To ascertain whether or not magnetic anisotropy plays a role, we adopted the approach, proposed by Joy $et~{}al.$ [26], according to which, if the decrease in M_ZFC at low temperatures is due to magnetocrystalline anisotropy, M_ZFC should follow the relation, $\frac{M_{FC}}{H_{app}+H_{c}}\approx\frac{M_{ZFC}}{H_{app}}$. That this is indeed the case for the fields $H\geq$ 100 Oe, is evident from Fig. 2(e - h). However, at $H\leq$ 50 Oe, $\chi_{ZFC}$ ($=\frac{M_{ZFC}}{H_{app}}$) strongly deviates from $\chi^{{}^{\prime}}_{FC}$ ($=\frac{M_{FC}}{H_{app}+H_{c}}$) (see Fig. 2(c, d)) in the low temperature region. The demagnetization-limited behavior of $\chi_{ZFC}$ at $T<T_{c}$ for $H=$ 10 Oe, suggests that the shape anisotropy also becomes important at low fields $H\leq$ 50 Oe. Note that, in the calculation of $\chi^{{}^{\prime}}_{FC}$, we have used the coercive field, H_c(T), data shown in the inset of Fig. 3.

Fig. 3 presents the M(H) hysteresis loops at fixed temperatures, taken in the ‘field-cooled’ (H = 70 kOe) mode at temperatures below and above the PM-FM transition temperature, T_c = 220.5 K, (corresponding to the dip in dM_ZFC(T)/dT at low fields H = 10 - 100 Oe, not shown here). With temperature decreasing from T $\gg$ T_c, where the sample is in the PM state and as such the M(H) isotherms are linear, the coercive field, H_c, increases from zero at T = T_c = 220.5 K to about 800 Oe at 5 K; the rate of increase in H_c picks up for temperatures below 30 K, as is evident from the inset of Fig. 3. A steep increase in H_c as the temperature falls below 30 K is indicative of an increase in the random magnetic anisotropy (RMA). The observation that magnetization does not saturate even in fields as high as 70 kOe at low temperatures (Fig.5(a)) and the presence of RMA (as inferred from the critical-point analysis in the following section) support the existence of a CSG state at low temperatures.

The panels (a) and (b) of Fig. 4 display the temperature variations of the real ($\chi^{\prime}$) and imaginary ($\chi^{\prime\prime}$) parts of ac magnetic susceptibility, respectively. While the inset (i) of panel (a) highlights the frequency-induced shift in the peak in $\chi^{\prime}$(T) at T = $T_{p}$ $\simeq$ 8 K, the inset (ii) clearly brings out the frequency-independence of the dip in the d$\chi^{\prime}$/dT $vs.$ T plots at $T_{c}$ = 220.5 K, which corresponds to the inflection point in $\chi^{\prime}$(T). The frequency-independent value of $T_{c}$ = 220.5 K confirms the true thermodynamic nature of the FM - PM phase transition at $T_{c}$.

The shift in $T_{p}$ per decade of frequency ($\Delta T_{p}/\{T_{p}\Delta(log_{10}\omega)\}$ = 3.2 $\times$ 10^-2) is an order of magnitude greater than that ($\simeq$ 10^-3)) observed [31] in canonical spin glasses but is typical of the cluster spin glasses [31, 36]. The frequency dependence of the peak temperature, $T_{p}(\omega)$, is analyzed in terms of the following expression given by the ‘critical slowing down’ model [31, 36]

where $\tau_{o}=2\pi/\omega_{0}$ is the relaxation time due to the correlated spin dynamics, $T_{g}$ is the spin-glass transition temperature in the zero-frequency limit, $\nu$ is the spin-spin correlation length critical exponent and $z$ is the dynamical critical exponent. The linear fit through the $\omega$ versus $({T_{p}(\omega)-T_{g}})/T_{g}$ data (solid circles) in the inset (iii) of Fig. 4, based on the Eq. (1) with the parameter values $\tau_{o}$ = 2.49(1) $\times$ 10^-8 s, $T_{g}$ = 7.68(2) K and the product $z\nu$ = 7.98(2), testifies to the validity of the critical slowing down model. Note that, in the canonical spin glasses, relaxation time, $\tau_{o}$, is typically of the order of 10^-12 s [31, 36]. Several orders of magnitude larger $\tau_{o}$ (i.e., much slower spin dynamics), in the present case, further confirms the existence of a cluster spin glass ground state.

Next, we focus our attention on the FM - PM transition at $T_{c}$ = 220.5 K. In the critical region above $T_{c}$, the inverse of measured ac susceptibility, $\chi_{ac}^{-1}$(T) $\equiv\chi^{\prime-1}$(T), is related to the inverse intrinsic susceptibility, $\chi_{int}^{-1}$(T) as $\chi_{ac}^{-1}$(T) = 4$\pi$ N + $\chi_{int}^{-1}$(T), where N is the demagnetizing factor. As the temperature approaches $T_{c}$ from above, $\chi_{int}$ diverges (or equivalently, $\chi_{int}^{-1}$ goes to zero) at $T_{c}$ in accordance with the relation

where $A$ and $\gamma$ are the critical amplitude and critical exponent, respectively. Inset (iv) of Fig. 4 depicts the best theoretical fit (continuous curve) to the $\chi_{ac}^{-1}(T)$ data, based on the Eq. (2), yields $T_{c}$ = 220.5 K, $A$ = 5.9(1) $\times$ 10⁵ and $\gamma$ = 1.261(5). This value of $\gamma$ will be put into a proper context later in the text.

It is well-known that the critical-point (CP) analysis of magnetization, M(T,H), data taken in the critical region near T_c provides a powerful means to unravel the true nature of magnetic ordering prevalent in a spin system. As elucidated below, the CP analysis makes use of the critical exponents $\beta$, $\gamma$ and $\delta$ for the spontaneous magnetization (order parameter), ‘zero-field’ susceptibility and the critical M(H) isotherm, (characterizing a continuous FM-PM phase transition at T_c), that are defined as  [37]

where $\epsilon=(T-T_{c})/T_{c}$, A, B and M₀ are the critical amplitudes [37, 38]. Diverse systems having exactly the same values for the critical exponents $\beta$, $\gamma$ and $\delta$, fall into a single universality class. Universality class, in turn, is governed by the order parameter (spin) dimensionality ($n$) and spatial dimensionality ($d$) [37, 39] as long as the interaction coupling the spins is of short range. The presence of different valence and spin states in the present system is expected to have a profound effect on the critical behavior. This expectation called for a detailed study of the critical behavior of LaSrCoO₄ at temperatures in the vicinity of the FM-PM phase transition. To this end, we have analyzed the virgin M(H) isotherms in the critical region (shown in Fig. 5(a)) using the scaling equation of state (SES) method, detailed in the reference  [37]. Instead of following the customary practice of using, at first, the Arrott [40] SES, we use the generalized form of the Arrott SES, given by Arrott and Noakes [41] (AN), i.e.,

to arrive at the correct choice of the critical exponents that makes the AN, $M^{1/\beta}$ versus $(H/M)^{1/\gamma}$, plots a set of parallel straight lines in the critical region near T_c. The temperature at which the linear AN plot isotherm passes through the origin marks the T_c. To begin with, we ascertain if the values of critical exponents $\beta$ and $\gamma$, theoretically predicted for any universality class, make the AN plot isotherms at temperatures close to T_c linear. For example, in Fig. 5 (b) it is evident that the mean-field (MF) critical exponent values $\beta$ = 0.5 and $\gamma$ = 1 do not yield linear AN isotherm in the critical region. The data presented in Fig. S1 of the supplementary material demonstrate that, like the MF critical exponent values, none of the universality class critical exponent choices: $\beta$ = 0.365, $\gamma$ = 1.386 for three-dimensional (3D) Heisenberg; $\beta$ = 0.345, $\gamma$ = 1.316 for 3D-XY; $\beta$ = 0.325, $\gamma$ = 1.241 for 3D Ising and $\beta$ = 0.25, $\gamma$ = 1.0 for mean-field tricritical, results in the linear AN plot isotherms.

Furthermore, regardless of the choice of the critical exponents $\beta$ and $\gamma$, extrapolation of the high-field portions of the AN isotherms to $H=0$ does not give any intercept on the $M^{1/\beta}$ axis. No intercept on the ordinate axis implies no spontaneous magnetization and hence absence of long-range FM order. This result prompted us to look for even the non-universal exponent values that could lead to the desired linear AN plot isotherms.

By varying the values of $\beta$ and $\gamma$ in Eq. (6), we succeeded in making the AN plot isotherms nearly straight over the field range 1.5 kOe $\lesssim H\lesssim$ 10 kOe with the critical exponent values, $\beta$ = 0.796, $\gamma$ = 1.051. Note that the field range for the linear AN isotherms is much wider in the immediate vicinity of T_c. From the modified Arrott plot (Fig. 6), it is evident that the extrapolation of the linear portions of the AN isotherms to H = 0 yields finite spontaneous magnetization, $M_{s}$, (inverse susceptibility, $\chi^{-1}$) below (above) T_c and both $M_{s}$ and $\chi^{-1}$ are zero at the transition temperature T_c $\simeq$ 220 K. This value of T_c is very close to that deduced from the ac susceptibility data. Using the critical exponent values, $\beta$ = 0.796 and $\gamma$ = 1.051, in the Widom scaling relation [42] $\delta=1+\gamma/\beta$, we obtained the critical isotherm critical exponent $\delta$ as 2.32. To crosscheck this value, the relation (Eq. (5)) is used to fit the virgin M(H) isotherm at T_c $\simeq$ 220 K. Fig. 7(c) shows the critical M(H) isotherm plotted as the ln M versus ln H_eff plot. As expected from Eq. (5), this log-log plot is linear with inverse slope $\delta$ = 2.35(5). This value of $\delta$ is in excellent agreement with that obtained using the Widom scaling relation.

Despite a close agreement between the value of $\delta$ directly determined from the critical isotherm and that from the exponent values $\beta$ = 0.796 and $\gamma$ = 1.051 via the Widom scaling relation, two basic questions arise: (i) how does one reconcile with the non-universal critical exponent values, and (ii) what causes the deviations from the linear AN isotherms at sufficiently low fields. Considering that the random magnetic anisotropy (RMA) model [43] yields nonlinear mean-field AN isotherms at low fields, it is worth investigating if the SES form given by Aharony and Pytte [43] helps resolve the above issues. Note that nonlinear mean-field isotherms have been observed in the past in a number of amorphous rare earth - transition metal systems and explained  [44] in terms of the Aharony and Pytte model.

Neglecting the critical fluctuations, Aharony and Pytte (AP) make the following predictions for a ($d$, $n$) spin system with RMA. (I) The M(H) isotherms follow the relation $H\sim M^{1/\delta}$ at $T_{c}$ and $H\sim M^{1/\delta_{1}}$ at all temperatures below $T_{c}$, with the exponents $\delta$ and $\delta_{1}$ given by

For a three-dimensional ($d$ = 3) system, Eq. (7) gives $\delta$ = 7/3 = 2.33 and $\delta_{1}$ = 5.0. While the theoretical estimate $\delta$ = 2.33 is in perfect agreement with the presently determined value $\delta$ = 2.35(5), the observed $\delta_{1}$ values do not reproduce the theoretically-predicted abrupt jump in $\delta_{1}$ at T $\simeq$ $T_{c}$ to the value $\delta_{1}$ = 5.0, which remains constant for T $<$ $T_{c}$ (as evidenced in Fig. 7(d)). This discrepancy between theory and experiment can be traced back to a total neglect of critical fluctuations of the order parameter, leading to Eq. (7). (II) According to the AP model, the scaling equation of state has the following form [43] for a $d$ = 3, $n$ = 3 system with random uniaxial and cubic anisotropies

where $\beta$ = 0.5, $\gamma$ = 1.0, $c\sim(D/J)^{2}$, $D$ and $\kappa$ are a measure of random uniaxial and cubic anisotropies, respectively, and $J$ signifies the exchange interaction. Eq. (8) permits a clear distinction between the cases: (i) $\kappa$ = 0 when only the random uniaxial anisotropy exists, and (ii) $\kappa$ $\neq$ 0 when, in addition to the random uniaxial anisotropy, cubic anisotropy is present. The low-field $M_{T}(H)$ data, plotted in the form of the Arrott plot (i.e., $M^{1/\beta}$ $vs.$ $(H/M)^{1/\gamma}$ plot with mean-field exponents $\beta=0.5$ and $\gamma=1$) isotherms, and the theoretical fits based on the AP SES, Eq. (8), for the cases $\kappa$ = 0 (dashed curves) and $\kappa$ $\neq$ 0 (continuous curves), are displayed in Fig. 7 (a). Fig. 7 (b) provides an enlarged view of the data and fits at fields not very far from zero. Evidently, the AP SES correctly reproduces the observed change in the curvature of the Arrott plot isotherms from concave-downward to concave-upward at T_c. Furthermore, superiority of the $\kappa$ $\neq$ 0 fits over those corresponding to $\kappa$ = 0, particularly at fields close to zero, asserts that the inclusion of cubic anisotropy is necessary. Consequently, in conformity with the observed behavior, an extrapolation to $H$ = 0 gives finite intercepts on the ordinate (finite M_s) for T $<$ T_c and on the abscissa (finite $\chi^{-1}$) for T $>$ T_c; both M_s and $\chi^{-1}$ go to zero at T = T_c. However, due to sizable gradients of the AP isotherms at very low fields ($H\rightarrow$ 0) for temperatures on either side of T_c, the intercepts on the ordinate and abscissa depend on the range of $H$ chosen for extrapolation. Thus, the M_s(T) and $\chi^{-1}$(T), so obtained, are not accurate enough for the determination of critical exponents $\beta$ and $\gamma$. In sharp contrast, the ‘zero-field’ ac susceptibility (ACS) does not suffer from such extrapolation errors. The value $\gamma$ = 1.261(5) of the susceptibility critical exponent, determined from the ACS data, is thus reliable. The deviation of this value of $\gamma$ from the mean-field value, $\gamma$ = 1.0, underscores the importance of critical fluctuations (neglected in the AP RMA model). Thus, the RMA model, due to Aharony and Pytte [43], captures the main physical essence in that the curvature in the mean-field Arrott isotherms at low fields has its origin in the random uniaxial and cubic anisotropies, but fails to yield correct values for the critical exponents $\beta$ and $\gamma$.

The evidence presented so far asserts that (i) at temperatures in range 180 K $\leq$ T $\leq$ T_c = 220.5 K, RMA is too weak to destroy long-range FM order, and (ii) the transition to a cluster spin glass (CSG) state occurs at T_g = 7.7 K. It remains to be verified if, in the intermediate temperature range, a mixed state is formed in which long-range FM order coexists with the CSG order. In this connection, Fig. 8(a) makes it obvious that, in the low temperature region (T $\lesssim$ 30 K), the Arrott (M² versus H/M) plots, when extrapolated to H = 0, do not yield any intercept on the ordinate axis regardless of the field-range chosen for extrapolation. It immediately follows that no spontaneous magnetization, and hence no long-range FM order, exists at these temperatures. Concurrently, in the same temperature range T $\lesssim$ 30 K, a steep rise in RMA (reflected in a sharp increase in the coercive field, as noticed in the inset of Fig. 3) occurs. Within the framework of the RMA models [43, 45], sizable magnitude of RMA at T $\lesssim$ 30 K can account for both the breakdown of long-range FM order [43] and the freezing of spin clusters in random orientations at T = T_g, leading to the formation of cluster (correlated) spin glass state (with a finite FM correlation length [45]) at temperatures below T_g.

With a view to gain more insight into the nature of magnetism at temperatures intermediate between the spin glass transition temperature T_g $\simeq$ 8 K and T_c $\simeq$ 220.5 K, the time (t) evolution of the ZFC magnetization, $M_{ZFC}$(t), at 150 K was measured in a static magnetic field of 50 Oe switched on after a fixed waiting time, $t_{w}$, ranging between $10^{2}$ and $10^{4}$ s. The $M_{ZFC}$(t) data, so obtained, were fitted to the stretched exponential function given by expression

where $M_{0}$ represents the FM component while $M_{r}$ and $t_{r}$ are the spin-glass magnetization component and the characteristic SG relaxation time, respectively [46]. The value of exponent $x$ reflects the type of energy barriers present. For instance, $x$ = 1 describes the relaxation behavior of a FM system with a single anisotropy energy barrier separating the ZFC and FC states  [47]. $x$ $\simeq$ 0.5, on the other hand, is associated with the stretched exponential relaxation, caused by hierarchical energy barriers in spin glasses [31]. From the fit (solid red curve through the data points), based on the above expression (Eq. (9)), to the $M(t)/M(0)$ data (Fig. 8(b)), we obtain $x=0.45(1)$, $M_{0}/M(0)$ = 1.083(1), $M_{r}/M(0)$ = 0.086(1) and $t_{r}$ = 2252 sec, where M(0) = 1.0348 is magnetization value, recorded at the time $t_{w}$ when the field was applied. The determined value of $x$ is in good agreement with those generally reported for spin glass systems [31]. This result provides an evidence for the coexistence of a glassy (a cluster spin glass-like) phase with ferromagnetic order at intermediate temperatures. Note that, within the uncertainty limits, the above-mentioned parameter values seem to be insensitive to the choice of $t_{w}$ presumably because $M_{r}$ constitutes a tiny fraction of the total magnetization at the measurement temperature of 150 K, which lies well above T_g $\simeq$ 8 K.

Finally, an attempt is made to ascertain how the present results can be understood in terms of the HS/IS/LS spin states of Co³⁺ ions in the LaSrCoO₄ compound. At first, we recall that the single-ion magneto-crystalline anisotropy (MCA) has its origin in the combined action of the crystal-field interaction (CFI) and spin-orbit coupling (SOC). Since the LS state is associated with S = 0, the LS Co³⁺ ions, neither in isolation nor as neighbors, contribute to spin magnetic moment and MCA (due to the lack of SOC). However, in the case of LS Co³⁺ ions with HS neighbors, the spin-state configurations, fluctuating  [12, 11] between the Co³⁺ HS - O^2- - Co³⁺ LS - O^2- - Co³⁺ HS and Co³⁺ IS - O^2- - Co³⁺ HS - O^2- - Co³⁺ IS configurations (refer to the schematic diagram shown in Fig. 9), can lead to FM superexchange interaction between the spins of Co³⁺ HS ions. This process involves correlated hopping [12, 11] of $t_{2g}$ and $e_{g}$ electrons between Co³⁺ LS and its Co³⁺ HS neighbors without any charge transfer as shown in Fig. 9 and results in a finite effective S at the LS Co³⁺ ion site. Finite S, in turn, enables SOC and ensures a significant contribution to spin magnetic moment. In conformity with the observed steep rise in MCA as the temperature drops below $\sim$ 80 K, MCA increases with decreasing temperature because LS Co³⁺ ions with SOC and much stronger CFI (and hence MCA) increase  [11, 12, 48] in number at the expense of HS Co³⁺ ions. In sharp contrast, arguments based on the mixed HS-IS or IS ground state (similar to those presented above), at best, lead to a much slower increase in MCA with lowering temperature because of their low CFI relative to LS. A competition between the above-mentioned FM interaction and the AFM interaction between the spins of neighboring Co³⁺ HS ions, caused by the Co³⁺ HS - O^2- - Co³⁺ HS superexchange, give rise to a spin glass (SG) ground state.

In the LaSrCoO₄ compound, as functions of temperature, ‘zero-field-cooled’ (ZFC) dc susceptibility, $\chi_{dc}=M/H$, at low fields (H $\leq$ 100 Oe), and ac susceptibility, $\chi_{ac}$, (ACS), at different frequencies of the driving ac field of rms amplitude $h_{ac}$ = 3 Oe, exhibit peaks at around 200 K and 8 K. While the low-temperature peak in $\chi_{ac}$(T) shifts to higher temperatures with increasing frequency, the inflection-point in $\chi_{ac}$(T) (or equivalently, a dip in $d\chi_{ac}/dT$) at 220.5 K is frequency-independent. Note that the dip in $d\chi_{dc}/dT$ also occurs at exactly the same temperature. This observation provides strong experimental evidence for the existence of a thermodynamic PM - FM phase transition at T_c = 220.5 K. The frequency-induced shift in the low-temperature peak is described well by the critical slowing down model for a cluster spin glass (CSG). According to this model, the transition to the CSG state takes place at T_g = 7.7 K in the zero-frequency limit. The Aharony-Pytte (AP) model for a (d = 3, n = 3) Heisenberg spin system with random magnetic anisotropy (RMA) yields the value ($\delta$ = 7/3) of the critical exponent for the critical isotherm at T_c = 220.5 K, which agrees closely with the observed value of $\delta$ = 2.35(5). Moreover, the AP scaling equation of state correctly describes the observed crossover from the concave-downward to concave-upwards curvature in the low-field Arrott plot isotherms, as the temperature is increased through T_c. These findings clearly establish that the PM-FM transition is basically driven by RMA.

For temperatures below $\approx$ 30 K, large enough RMA destroys long-range FM order by breaking up the infinite FM network into FM clusters of finite size and leads to the formation of a CSG state at temperatures T $\lesssim$ 8 K by promoting freezing of finite FM clusters in random orientations. Increasing strength of the single-ion magnetocrystalline anisotropy (and hence RMA) with decreasing temperature reflects an increase in the number of low-spin (LS) Co³⁺ ions at the expense of that of high-spin (HS) Co³⁺ ions. At intermediate temperatures (30 K $\lesssim T\lesssim$ 180 K), spin dynamics has contributions from the infinite FM network (fast relaxation governed by a single anisotropy energy barrier) and finite FM clusters (extremely slow stretched exponential relaxation due to hierarchical energy barriers).

The authors acknowledge Kranti Sharma for AC susceptibility measurements. S.S.M. acknowledges the financial support from SERB, India, in the form of the national post doctoral fellowship (NPDF) award (PDF/2021/002137). R.S. acknowledge the financial support provided by the Ministry of Science and Technology in Taiwan under project numbers MOST-111-2124-M-001-009 & MOST-110-2112-M-001-065-MY3 & AS- iMATE-111-12.