Spin-liquid state with precursor ferromagnetic clusters interacting antiferromagnetically in frustrated glassy tetragonal spinel Zn_0.8Cu_0.2FeMnO₄

The structural characterization of the prepared sample of Zn_0.8Cu_0.2FeMnO₄ using x-ray and ND confirmed the monophasic nature of the sample. As evident in the x-ray pattern (figure 1) collected at RT (295 K), this compound exhibits the tetragonal spinel structure with lattice parameters a= 5.8390(7) Å and c = 8.8156(14) Å giving a c/a√2 ratio of 1.0676(3). For comparison, the parameter a= 8.52 Å was found for cubic ZnFe₂O₄ [10], and a= 5.72 Å and c = 9.245 Å has been reported for tetragonal ZnMn₂O₄ giving a c/a√2 ratio of 1.143 [20]. The elongation along the c direction results from the presence of JT active Mn³⁺ ions. However, this JT effect is also well established in Zn_0.8Cu_0.2FeMnO₄, where the tetragonal distortion is smaller and therefore weaker pronounced than in ZnMn₂O₄. From our neutron data collected at 1.7 K it is found that the tetragonal splitting is slightly smaller than that obtained at RT leading to c/a√2 = 1.06054(10). Here, it is noted that no changes could be detected for Zn_0.8Cu_0.2FeMnO₄ concerning the structural part between 1.7 K and 79.4 K. Further, our refinements revealed anisotropic peak broadenings, especially for the reflections (220) and (004). The fit could be improved by using strain parameter implemented in the FullProf suite [21]. Due to the better instrumental resolution of the x-ray data, the half widths could be determined more precisely. Interestingly, the best fit of the integrated intensities (shown in the inset in figure 1) could be obtained by using a Lorentzian peak shape.

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To obtain more information on the crystal structure properties we carried out Rietveld refinements of the powder ND pattern recorded at 1.7 K. Here, it is noted that the scattering power of oxygen is more pronounced in ND than in XRD which allowed us to determine its positional parameters with higher accuracy. Further, we can get important information on the distribution of the metal atoms Mn, Fe, Cu and Zn in the AO₄ tetrahedra and BO₆ octahedra since their neutron scattering lengths show considerable differences, whereas their x-ray form factors are very similar. A satisfactory fit can be obtained for refinements in the tetragonal space group I4₁/amd (No. 141, cell choice 2). The metal ions at the A and B sites are located at the Wyckoff sites 4b(0,¼,⅜) and 8c(0,0,0), respectively, and the O atom at the site 16h(0,y,z). The refinements resulted in a residual R_F = 0.0155, where the occupancies of the A and B sites are initially allowed to vary using the constraints occ(Cu_A ) + occ(Zn_A ) = 1 and occ(Fe_B ) + occ(Mn_B ). Here it has to be noted that the occupancies of the B site atoms Fe and Mn can be determined with much higher accuracy because of their strongly different neutron scattering lengths resulting in the values occ(Fe_B ) = 0.4986(10) and occ(Mn_B ) = 0.5014(10) which is very close to ratio of 1:1. Nonetheless, in the case of refinements of the occupancies at A site the difference of neutron scattering lengths of Cu and Zn are much smaller, resulting consequently in an enlarged standard deviation. This may also be the reasons that the occupancies are not in good agreement with the expected values as per the chemical stoichiometry, because of the dilute incorporation of Cu²⁺, where for occ(Zn_A ) and occ(Cu_A ) the values 0.913(11) and 0.087(11) (instead of 0.8 and 0.2) are found, respectively. Therefore, we have inspected the possibility that Cu at the A site is fully or partially replaced either by Fe or Mn. In the next step we carried out a refinement, where Cu is partially replaced by Mn which has a negative neutron scattering length. In this refinement we have set occ(Zn_A ) = 0.8 and occ(Fe_B ) = 0.5 as well as the constraints 2 × occ(Cu_B ) = occ(Mn_A ) and occ(Cu_A ) + occ(Mn_A ) = 0.2. Finally, we obtained the stoichiometry (Mn_0.0128(18)Cu_0.1872(18)Zn_0.8)_A [FeMn_0.9872(18)Cu_0.0128(18)]_B O₄. This would only give a replacement of Cu by Mn at the A site of 6.4%, which does not support existence of Mn at the A site. At this point feasibility of either of the chemical formula (Zn_0.8Cu_0.2)_A [FeMn]_B O₄ and (Zn_0.8Fe_0.2)_A [Cu_0.2Fe_0.8Mn]_B O₄ cannot be ruled out. If both Cu and Fe occupy the A site along with Zn, the following constraints are used: occ(Zn_A ) = 0.8, occ(Mn_B ) = 0.5, 2 × occ(Cu_B ) = occ(Fe_A ) and occ(Cu_A ) + occ(Fe_A ) = 0.2. Finally, the refinements result in the occupancies occ(Cu_A ) = 0.214(4) and occ(Fe_A ) = −0.014. This is a strong indication that the A site is fully occupied with Zn and Cu whereas the B site with Fe and Mn giving the chemical formula (Zn_0.8Cu_0.2)_A [FeMn]_B O₄.

The refinement of the structural parameters of Zn_0.8Cu_0.2FeMnO₄ (table 1) revealed a strong elongation of the T_B O₆ octahedra, where the apical bond length d_ab (T_B –O) = 2.1274(11) Å is somewhat larger than the equatorial one d_c (T_B –O) = 1.9752(7) Å which is finally the cause of the increase of the c/a√2. A similar trend was found recently for mixed spinel Ti_0.6Mn_0.4Co₂O₄, but weaker pronounced, having the bond lengths d_c (T_B –O) = 2.025(4) Å and d_ab (T_B –O) = 1.998(2) Å despite the fact that the c/a√2 ratio is very close to 1 [23]. The enlarged c/a√2 of Zn_0.8Cu_0.2FeMnO₄ also leads to an elongation of the T_A O₄ tetrahedra. Here the bond angle ∠_c (O–A–O) = 107.76(8)°, having a bisector along the c axis, is slightly smaller than the ideal tetrahedron angle of 109.47°. More information about cation distribution and valence states of the determined interatomic T–O distances of Zn_0.8Cu_0.2FeMnO₄ are compared with those obtained from the ionic radii given by Shannon [24]. It is noted that the ionic radius of Cu²⁺ (r_Cu2+ = 0.57 Å) is closer to the value of Zn²⁺ (r_Zn2+ = 0.6 Å) than to that of Fe³⁺ (r_Fe3+ = 0.49 Å). If one assumes that iron occupies the A site instead of Cu²⁺ it is more likely that iron is in the valence state 2+ (r_Fe2+ = 0.63 Å, high spin). With this insight, the bond length in the T_A O₄ tetrahedron is found to be d(T_A –O) = 1.9667(10) Å. This value is somewhat smaller than the value 2.01 Å using the ionic radii of Shannon if the composition at the A site is Zn²⁺ _0.8Fe²⁺ _0.2. If the composition Zn²⁺ _0.8Cu²⁺ _0.2 (Zn²⁺ _0.8Fe³⁺ _0.2) is assumed the value 1.95 Å (1.87 Å) is obtained using the ionic radii of Shannon. Here, it is noted that the value 1.95 Å of the composition Zn²⁺ _0.8Cu²⁺ _0.2 is only slightly smaller than the refined bond distance d(T_A –O) = 1.9667(10) Å. All these aspects again strongly indicate the chemical formula is simply (Zn_0.8Cu_0.2)_A [FeMn]_B O₄ for which the cationic and spin states of all the individual atoms are determined below.

We also compare the determined bond distance in the T_B O₆ octahedron with that obtained with the ionic radii given by Shannon. Assuming a high- or low-spin state of both Mn³⁺ and Fe³⁺ one obtains the values 2.045 Å and 1.965 Å, respectively. In our study, we obtained the bond lengths d_ab (T_B –O) and d_c (T_B –O) (see table 1) giving an averaged value of 2.026 Å that is closer to the value given above for the high-spin states. Interestingly, the Fe³⁺ ions in ZnFe₂O₄ carry at the B site a magnetic moment in the ordered state of 3.9 µ_B [25]. In Mn₃O₄, the ordered moment at this site is µ_tot = 3.55 µ_B [26]. For Zn_0.8Cu_0.2FeMnO₄ no LRO occurs at 1.7 K as shown in the neutron powder diffraction pattern (figure 1), where the magnetic reflection is strongly broadened indicating the presence of a SG state. Using the moment values of ZnFe₂O₄ and Mn₃O₄ we can estimate for Zn_0.8Cu_0.2FeMnO₄ a magnetic moment of 3.7 µ_B at the B site, which refers to a combination of both high- and low-spin states on the octahedral site.

To further confirm the electronic states and probe the elemental stoichiometry in detail, we performed the XPS measurements. The recorded XPS spectra for all the individual elements present in ZCFMO are plotted as a function of binding energy (in eV) as displayed in figure 2. Here the adventitious carbon (C-1s) peak at 284.8 eV was used as a charge reference to standardize the core level spectra of all the individual atoms namely: Cu-2p, Zn-2p, Fe-2p, O-1s and Mn-2p. Figure 2(a) shows the core level spectrum of Zn-2p which exhibits two sharp cusps centered across 1021 eV (2p_3/2) and 1044 eV (2p_1/2) with no signatures of additional satellite peaks. The energy difference between these two cusps (ΔZn-2p_(1/2−3/2) ∼ 23 eV) confirms the divalent character of Zn in ZCFMO [27, 28]. Further, two humps are noticed in the core level XPS spectrum of Mn-2p (figure 2(b)) around 641.2 eV and 653.25 eV, associated with the 2p_3/2 and 2p_1/2, respectively, confirms the trivalent electronic state of Mn [23]. On the other hand, the XPS spectra of Fe-2p, depicted in figure 2(c), consists of at least four main peaks, which are deconvoluted into six number of peaks situated at 710.75 eV (F₁), 712.9 eV (F₂), 718.85 eV (F₃), 725.1 eV (F₄), 714.65 eV (S₁) and 733.1 eV (S₂). Here S₁ and S₂ are the satellite peaks corresponding to the shoulders noticed across the maximum intensity peaks of lower binding energies. The peak centered at F₁ is associated with the 2p_3/2 core level spectra of divalent Fe, while the peak located across F₂ belongs to Fe³⁺ [29, 30]. Additionally, the spin–orbit splitting energy for these maxima are calculated to be Δ(F₃ ‒ F₁) ∼ 8.1 eV and Δ(F₄ ‒ F₂) ∼ 12.2 eV, which again confirms the mixed-valent character of Fe present in the ZCFMO system. The main panel data of figure 2(d) possesses a sharp peak across 529.75 eV pertaining to the O-1s spectra of ZCFMO. Whereas, the inset in figure 2(d) represents the XPS spectra of Cu which is comparatively weaker because of their dilute substitution in the spinel lattice. Nonetheless, we notice that the Cu-2p core level spectra has been deconvoluted into total four number of peaks: C₁, C₂, C₃ and C₄, in which the main cusps C₁ and C₂ are located at 933.45 eV and 953.6 eV, respectively, while the other two are satellite peaks situated at C₃ ∼ 943.05 eV and C₄ ∼ 961.75 eV. The spin-orbit splitting energy (ΔC) between the two main peaks (C₁ and C₂) is obtained to be 20.15 eV confirms the presence of Cu²⁺, which is further supported by the broad satellite peak across 943.05 eV [31, 32]. In summary, the results from the analysis of XRD and XPS lead to the chemical formula for Zn_0.8Cu_0.2FeMnO₄ as (${\text{Zn}}_{0.8}^{2 + }\,{\text{Cu}}_{0.2}^{2 + }$)_A [${\text{Fe}}_{0.4}^{2 + }\,{\text{Fe}}_{0.6}^{3 + }\,{\text{M}}{{\text{n}}^{3 + }}$]_B O_4‒δ , with δ = 0.2 for reason of charge balance; for brevity, this is listed simply as ZCFMO in the rest of the paper.

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The temperature dependence of ac susceptibilities, χ' and χ'', of ZCFMO were measured at several frequencies f = 0.17–510 Hz with magnitude of ac field h_ac = 4 Oe and the dc field H = 0. For these measurements, the sample was cooled under H = 0 from 300 K to 4 K, known as the zero field cooled (ZFC) condition, followed by measuring χ' and χ'' with increasing temperature. In figure 3, results for a fixed frequency f = 17.13 Hz are presented which show only a single broad peak near 60 K for χ' vs. T whereas for χ'', there is a sharp peak near 47 K, with associated weaker broader peak near 63 K and a weaker peak near 9 K. In figure 4, temperature dependence of χ' and χ'' at various frequencies shows that with increase in frequency, the peak position of the 47 K peak shifts to higher temperatures.

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The temperature dependence of the peak near 9 K covering the temperature range from 4 K to 15 K for eight frequencies between 0.17 Hz and 510 Hz is shown in figure 5. From these results, the peak in the χ'' vs. T data is evident near 9 K, its peak position shifting to higher temperatures with increase in frequency. For the χ' vs. T data, there is a change in slope near 9 K, especially for higher frequencies.

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To determine the origin of these peaks near 47 K and 9 K, the Mydosh parameter Ω given by [33, 34]

$$\begin{equation}\Omega = ({T_{f\,2}} - {T_{f\,1}})/{T_{f\,1}}\left( {\log {f_2} - \log {f_1}} \right)\end{equation} \tag{ 1 }$$

is determined from the χ'' data using f₁ = 0.17 Hz and f₂ = 510 Hz with T_f being the position of the peak. For the peak near 47 K, this analysis gives Ω = 0.0246, and for the peak near 9 K, similar analysis gives Ω = 0.0448. These magnitudes are considerably larger than Ω = 0.005 observed in canonical SGs but close to the magnitudes for cluster SGs [35–37], especially for the peak near 47 K. So, this peak near 47 K is determined to be due to cluster SG transition. Additional discussion on the nature of the peak near 9 K is presented later.

Another approach to verify the nature of SG transitions is to do scaling analysis using the Vogel–Fulcher law (VFL) given by equation (2) and the power law (PL) given by equation (3) [35],

$$\begin{equation}\tau = {\tau _0}\exp [{E_{\text{a}}}/{k_{\text{B}}}({T_f} - {T_0})]\end{equation} \tag{ 2 }$$

$$\begin{equation}\tau = {\tau _0}{[({T_f} - {T_{{\text{SG}}}})/{T_{{\text{SG}}}}]^{ - z\nu }}.\end{equation} \tag{ 3 }$$

In these equations, τ =1/(2πf) is determined from the positions of the peaks in the χ'' vs. T at different f, E_a is the activation energy, T₀ represents the strength of inter-cluster interaction, T_SG is the SG transition and zν is the dynamical critical exponent [38]. From equation (2), plot of Ln τ vs. 1/(T_f − T₀) should yield a straight line with intercept Ln τ₀ and slope E_a/k_B. Similarly, from equation (3), plot of Ln τ vs. Ln [(T_f ‒ T_SG)/T_SG] should be linear with slope yielding zν. However, the optimum of T₀ and T_SG for such analysis is still not obvious.

In a recent paper from our group [39], we have presented a procedure to determine optimum magnitudes of T₀ and T_SG using an adjusted R² method, noting that adjusted R² = 1 for optimum values. In figure 6, we show the results of this procedure for the peak near 47 K by plotting adjusted R² vs. different choices of T_SG and T₀ and determining the corresponding magnitudes of τ₀, zν and E_a/k_B. This analysis shows that the optimum T_SG = 41.8 K with τ₀ = (2.05 ± 0.34) × 10^‒11 s and zν = 9.75 ± 0.08 for the fit to the PL and T₀ = 36.8 K, τ₀ = (6.0 ± 0.9) × 10^‒11 s and E_a/k_B = (195.1 ± 1.5) K for the fit to the VFL (figure 7). Since k_B T₀/E_a = 36.8/195.1 ≃ 0.19 is much less than unity, it shows that inter-cluster interaction is weaker than the energy barrier between clusters, leading to the cluster glass state without long range magnetic order. The magnitude of τ₀ and zν are characteristic of SGs, although it is now realized that their magnitudes cannot be used to distinguish between cluster and canonical SGs [31, 39]. In summary, the magnitude of Ω = 0.0246, combined with k_B T₀/E_a ≪ 1 establishes this system as a cluster SG. The reason for the difference of T_SG ≃ 41.8 K obtained from this procedure from 47 K is not yet clear. Our attempts to do this analysis using T_SG near 47 K were not successful.

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The shift in the peak position of χ'' at 47 K with applied dc field H = 20, 50, 100, 200, 300 and 800 Oe is shown in figure 8(a). This shift is analyzed with the help of non-mean field theory (de Almeida–Thouless line (AT-line) analysis) described by the equation: ${T_{{\text{SG}}}}\left( H \right) = {T_{{\text{SG}}}}\left( 0 \right)\left[ {1 - A{H^{2/\phi }}} \right]$ where T_SG(H) is the position of the peak in χ'' vs. T data in applied dc field H. The exponent φ= 3 for the AT-line pertains to the Ising system. Our fit in figure 8(b) yields φ= 3.37 not too different from the Ising case. In this plot, we have also plotted the shift in T_SG(H) vs. H (figure 8(b)) determined from the inflection point in the plot of ∂χ/∂T vs. T where χ (=M/H) is the dc susceptibility measured in presence of different magnetic fields H. The details of these measurements and analysis are presented in the next section. It is noted that the broad peak near 63 K present in low H values in figure 8(a) effectively disappears in larger H values. So, this peak is likely due to some domain structure and no additional discussion is presented on this peak. Another notable point from the variation of T_SG vs. H in figure 8(b) is that for H = 800 Oe, T_SG is lowered to about 25 K. Additional discussion on this is presented in the next section on the temperature and magnetic field dependence of magnetization measured under various applied magnetic fields H.

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The temperature dependence of the measured dc magnetic susceptibility (χ) for H = 500 Oe (up to 300 K) and H = 1 kOe (up to 400 K) is shown in figure 9. For T above ∼125 K, the two measurements coincide but for T < 125 K, χ at H = 1 kOe is noticeably lower than that at H = 500 Oe. Both data show peak near 60 K and a second inflection near 25 K. The χ vs. T data was fit to the modified Curie–Weiss (MCW) law: χ = χ₀ + C/(T ‒ θ), where χ₀ = ‒1.14 × 10^‒4 emu mol^‒1 Oe^‒1 was determined as the diamagnetic contribution based on the Hartree–Fock theory [40]. The plot of 1/(χ ‒ χ₀) vs. T is shown in figure 10. The data fits the MCW law for T > 250 K quite well with significant deviations observed for T < 250 K. The fit yields C = 3.29 emu K mol^‒1 Oe^‒1 and θ ≃ 185 K. The positive θ signifies that the dominant exchange interaction in the system is FM. However, the magnitude of 1/(χ ‒ χ₀) being higher than the MCW fit for T < 250 K signifies the presence of AFM interaction between the FM clusters. The magnitude of C = 3.29 emu K mol^‒1 Oe^‒1 is used to determine ${\mu _{{\text{eff}}}} = \left( {\sqrt {7.993C} } \right){\mu _{\text{B}}}$ yielding µ_eff = 5.13 µ_B. Using this value of µ_eff and the relation µ_eff/µ_B = g [S(S +1)]^1/2, yields effective spin S = 2.11 for g = 2 for this system. This magnitude of µ_eff can be explained if the B site Fe³⁺ and Mn³⁺ ions are in the low spin configuration with µ = 1.73 µ_B for Fe³⁺ and µ = 2.828 µ_B for Mn³⁺ ions with effective spins S = 1/2 and S = 1, respectively. Fe²⁺ in the high spin state has S = 2. This yields (µ_eff/µ_B)² = 0.2(1.9)² + 0.4(4.9)² + 0.6(1.73)² + (2.828)² = 20.12 or (µ_eff/µ_B) = 4.49. Here the values of spin-only moments are used, and so this may be the reason that the resultant theoretical moment (µ_eff≃ 4.5 µ_B) is slightly smaller than the moment µ_eff = 5.13 µ_B obtained from our susceptibility measurement. This also yields the B site moment µ ≃ 4.4 µ_B compared with µ ≃ 3.7 µ_B determined from the ND results discussed above.

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The second test on the correctness of this ionic arrangement is the measured M vs. H at 2 K, shown in figure 11, and the estimated saturation magnetization M_S ≃ 14000 emu mol⁻¹ = 2.55 µ_B/f.u. The measured moment for each ion is gS_Z . We take g = 2.2 for Cu²⁺ and g = 2 for Mn³⁺, Fe²⁺ and Fe³⁺ ions. This calculation yields µ = 2.78 µ_B for the arrangement (Cu²⁺↓)_A [Fe²⁺↑, Fe³⁺↓, Mn³⁺↑]_B close to the experimental value. Except for low-spin Fe³⁺, all the other magnetic ions located on the B site interact with dominant FM(↑) coupling, while the dilute Cu²⁺ on the A site couples antiferromagnetically. For µ_eff = 5.13 µ_B, effective S = 2.11 and using g = 2 should yield M_S = gSµ_B = 4.22 µ_B, if all the moments are aligned parallel to each other. Our experimental value of M_S ≃ 2.55 µ_B/f.u. indicates that all the moments are not aligned parallel to each other. Thus, this system is expected to have high degree of magnetic frustration as in the spinels ZnFe₂O₄ [8–11] and MgMnO₃ [18], which also have magnetic ions only on the B site. The above analysis shows that ferromagnetically aligned spins of Fe²⁺ and Mn³⁺ have AFM interaction with the antiferromagnetically aligned spins of Fe³⁺ on the B site and dilute Cu²⁺ spins on the A site although the effect of Cu²⁺ is likely insignificant because of its low concentration. So, this system can be considered to have ferromagnetically aligned spin clusters mediated by antiferromagnetically aligned spins. With this insight, our measurements reported here on the ac and dc magnetic susceptibilities, ND and temperature dependence of the specific heat are interpreted in terms of SRO of FM clusters interacting antiferromagnetically but without the presence of long-range magnetic ordering. These considerations lead to complex magnetic ordering in ZCFMO including SG state near 47 K and SL state at lower temperatures, the latter discussed in more details later.

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The temperature dependence of χ with H = 500 Oe and under the field cooled (FC) and ZFC protocols is shown in figure 12(a), clearly showing bifurcation of the two plots for T < 24 K along with a maximum across 60 K. It is known that in an antiferromagnet, the peak in χ vs. T does not represent magnetic ordering. Instead, it is the peak in ∂(χT)/∂T vs. T which corresponds to the Néel temperature as confirmed through peak in the specific heat versus temperature data [41, 42]. In ferromagnets, the negative peak in ∂χ/∂T vs. T is usually used as the magnetic ordering temperature [43, 44]. Because of the presence of both FM and AFM couplings in ZCFMO, the plots of both ∂(χT)/∂T vs. T and ∂χ/∂T vs. T are shown in figures 12(b) and (c), respectively. In figure 12(b), the peaks in ∂(χT)/∂T vs. T are observed at 97 K (negative peak), 47 K and 24 K whereas the positions of these peaks in ∂χ/∂T vs. T in figure 12(c) are shifted to lower temperatures by a few degrees. The peak near 97 K is associated with local FM ordering in clusters since its position shifts to higher temperatures with increase in applied H as expected in ferromagnets and ferrimagnets [45, 46]. Though the peak near 47 K coincides with the SG transition T_SG observed in ac susceptibilities discussed in detail in section 3.2, the maximum at 60 K in figure 12(a) likely appears due to the crossover dynamics of domains owing to the rotational processes or wall displacements in ferrimagnets/ferromagnets [47, 48]. The peak near 24 K below which there is a bifurcation of χ(FC) and χ(ZFC) plots, is associated with the shift of T_SG with applied H = 500 Oe as shown previously in figure 8(b). Interestingly, if we consider the local minima or the inflection point between the peaks 24 K and 47 K as T_SG, which is obtained at 31 K based on the ∂χ/∂T vs. T plot, the results perfectly agree with the ac susceptibility results in figure 8(b).

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To understand the transition near 24 K below which χ (FC) > χ(ZFC) in figure 12(a), we measured the temperature dependence of hysteresis loop parameters. The temperature dependence of the coercivity H_C in the inset of figure 13 shows that for T > 24 K, H_C goes to zero. This suggests that for T < 24 K, the system develops a partially ordered state in a magnetic field but without signatures of exchange-bias or loop shift like the situation in magnetically diluted system like Co_x Mg_1‒x O [49].

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As noted earlier, the dominant exchange constant which governs the high temperature χ vs. T variation is FM. To determine this dominant exchange interaction, we followed the high temperature series expansion (HTSE) method, based on fitting the paramagnetic χ_p vs. T, prescribed for spinel structure following Heisenberg Hamiltonian [31, 50, 51]. This HTSE expression is given below:

$$\begin{equation}{\chi _p} = \frac{C}{T}\sum\limits_{i = 0}^6 {C_i}{\left( {\frac{{ - J}}{{{k_{\text{B}}}T}}} \right)^i}.\end{equation} \tag{ 4 }$$

Here, k_B is the Boltzmann's constant, χ_p = χ ‒ χ_0, C_i 's are the normalized coefficients calculated up to 6^th order as suggested by Schmidt et al using the parameter 'r = $\mu _{{\text{eff}}}^2/4$' in which µ_eff = 5.13 µ_B is used as determined previously from the fit to MCW law. The coefficients obtained using this procedure are: C₀ = 1, C₁ = ‒13.2, C₂ = 141.25, C₃ = ‒1333.47, C₄ = 11872.9, C₅ = ‒104854.14, C₆ = 934662. The fits of the χ vs. T data to equation (4) along with the fit to the MCW expression is shown in figure 14. The fit to HTSE represent only marginal improvement over the MCW fit and yield FM J/k_B = 17 K. It is noted that retaining just the first two terms (C₀ and C₁) of the series yield the approximate MCW law and J/k_B = 14 K using θ = 185 K. The departure of the fit from the experimental values at lower temperatures signifies the presence of AFM exchange interactions among FM clusters which are not considered in the above analysis because of the complexity of such calculations.

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To investigate the long-range vs. shortrange magnetic ordering in ZCFMO, the thermal variation of its heat capacity C_p was measured from 2 K to 200 K under ZFC conditions in applied H = 0 and 90 kOe. This variation of C_p vs. T is shown in figure 15 along with the plot of ‒ΔC_p = C_p (0) − C_p (90 kOe) vs. T. Although the magnitude of ‒ΔC_p is comparatively small, it shows that applied H suppresses C_p with a peak in its magnitudes around 80 K. This variation is likely due to coupling of the FM clusters to the applied H. More importantly, in the C_p (0) vs. T data, a λ-type anomaly characteristic of LRO is not observed as often reported for second order magnetic transitions [39, 52]. However, lattice contribution to C_p must be first subtracted out from the measured C_p before any magnetic contribution to C_p can be ruled out. The Debye model of lattice specific heat C_L which usually work quite well at lower temperatures yields the following expression for C_L :

$$\begin{equation}C_L{ } = { }9{ }R{ }\left( {T{ }/{{\Theta_D}}} \right)^3 \mathop \int \limits_0^{\frac{{{\Theta _D}}}{T}} \frac{{{x^4}{e^x}}}{{{{\left( {{e^x} - 1} \right)}^2}}}{\text{d}}\,x\end{equation} \tag{ 5 }$$

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where Θ_D is the Debye temperature, usually determined from the fit to the data and R = 8.314 J mol^‒1K^‒1 is the gas constant. For low temperatures when T ≪ Θ_D, so that the limit Θ_D/T in the integral in equation (5) can be replaced by infinity, C_L = 216 R (T/Θ_D)³ is obtained for the lattice contribution. Therefore, in the case of only lattice contribution to C_p, C_p /T³ should yield a constant value at lower temperatures. In the inset of figure 15, the plot of C_p /T³ vs. T up to 45 K shows that below about 25 K, there is indeed some magnetic contribution to C_p, whereas for higher temperatures C_p tends to become constant. This also shows that for T > 45 K, a good fit of the C_p vs. T to equation (5) can be expected for a specific choice of Θ_D. This fit of the C_p vs. T data for H = 0 Oe to equation (5) for Θ_D = (376.2 ± 0.4) K is shown in figure 16 along with the temperature variation of the evaluated C_M = C_p − C_L , where C_M is the magnetic contribution. The temperature variation of C_M has a broad peak around 40 K which coincides with T_SG determined from the analysis of the ac susceptibilities. This analysis confirms the presence of short range magnetic order associated with the cluster SG state.

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The variation of C_M vs. T is used to determine the change in temperature dependence of the magnetic entropy:

$$\begin{equation}{S_M} = \mathop \int \limits_0^T \left( {{C_M}/T} \right)\,{\text{d}}T.\end{equation} \tag{ 6 }$$

In the inset of figure 16, the thermal variation of C_M /T and computed S_M are shown. At 100 K, S_M ≃ 13.1 J mol^‒1K^‒1 which effectively remains constant above 100 K. But with decrease in temperature, S_M begins to decrease due to the onset of short-range magnetic ordering and S_M approaches zero below 4 K. Using S_M = R Ln(2S+ 1) = 13.1 yields effective S = 1.92 to be compared with S = 2.11 determined earlier from the analysis of paramagnetic susceptibility. Our system ZCFMO contains magnetic ions with S = 1/2, 1, and 2. So these composite values of S obtained from the analysis of magnetic susceptibility and specific heat are quite reasonable.

The temperature variation of C_p at liquid helium temperatures has been used to distinguish between SG and SL states in magnetically diluted systems. In canonical glasses which contain only a few percent of magnetic ions, C_p is known to vary linearly with temperature [53]. However, in other diluted systems with magnetic frustration such as SrCr₈Ga₄O₁₉ with Kagomé lattice [54, 55], NiGa₂S₄ with triangular lattice [56], CuCr_0.7V_0.3S₂ [57], and even in Co based glassy systems Co₂TiO₄ and Co₂SnO₄ [58], C_p varying as T² at low temperatures has been reported. This T² variation of C_p is considered as a signature of SL state in which magnetic moments fluctuate dynamically at very low temperatures [53–58]. For the ZCFMO system, the variation of C_p = AT ⁿ is tested by plotting log C_p vs. log T in figure 17 for the data taken for both H = 0 and H = 90 kOe. For T < 9 K, the data fits straight line with n = 1.83 for H = 0 and n = 1.95 for H = 90 kOe, the slope becoming smaller at higher temperatures. This analysis suggests SG state for T > 9 K changes to SL like state for T < 9 K. The ND studies presented earlier in this paper provide additional support for the dynamical fluctuations of the magnetic moment for T < 9 K. In the ac susceptibility measurements shown earlier in figures 3 and 5, there is a clear peak near 9 K in the plot of χ'' vs. T. These results provide sufficient proof for the transition to different magnetic phase below 9 K which has known signatures of a SL state.

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Additional information on the nature of magnetic ordering can be obtained by evaluating the change in magnetic entropy ΔS_M = S(H) − S(0) with change in magnetic field H. Since increase (decrease) in entropy represents increase (decrease) in disorder, ΔS_M is negative (and so −ΔS_M is positive) in the paramagnetic, FM and ferrimagnetic states as applied H improves spin ordering and hence lowers entropy. The opposite is valid for the AFM and SG states. ΔS_M can be determined from the M vs. H isotherms such as the one shown in figure 11 at 2 K using the Maxwell's equation [50]:

$$\begin{align}\Delta {S_M} = \mathop \int \limits_0^H {\left( {\frac{{\partial M}}{{\partial T}}} \right)_H}\,{\text{d}}H^{^{\prime}} = \frac{\partial }{{\partial T}}{\mathop \int \limits_{0}^{H}} M{\text{d}}H^{^{\prime}}\end{align} \tag{ 7 }$$

M vs. H isotherms were measured at many temperatures between 2 K and 200 K and the plot of ‒(ΔS_M ) vs. T determined from these plots using second part of equation (7) is shown in figure 18. This analysis is also used to determine the magnetocaloric effect (MCE) in materials. As seen from figure 18, a crossover from positive (normal MCE) to negative (inverse MCE) value of ‒ΔS_M is noticed across T_SG, since with increase in H, magnetic disorder decreases in the paramagnetic region but increases in the SG regime. A further drop is observed below the SL transition T_SL = 9 K. This analysis provides additional support for the SL state below 9 K.

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The time dependent behavior of frustrated systems in the SL region has rarely been reported in the literature. So, a detailed analysis of the thermo-remanent magnetization (TRM) studies on the ZCFMO system in the SL region is provided below, and it is compared with that reported in literature for the SG region. For this purpose, the system was cooled to the required low temperature under H = 50 kOe, H was then set to zero and the magnetization (M_TRM) was measured for a time (t) duration of 7600 s. The data is represented in figure 19 for the M_TRM vs. T isotherm obtained at T = 5 K. The main panel data is analyzed by employing several typical scaling laws and/or power laws such as [59, 60]:

$$\begin{equation}{M_{{\text{TRM}}}} = {M_0}-S\,Lnt \qquad\qquad\quad\;\;\;\,\end{equation} \tag{ 8 }$$

$$\begin{equation}{M_{{\text{TRM}}}} = {M_0}{t^{ - \gamma }}\qquad\qquad\qquad\quad\;\;\end{equation} \tag{ 9 }$$

$$\begin{equation}{M_{{\text{TRM}}}} = {M_0}\,\exp \left[ - {\left(\frac{t}{\tau }\right)^\alpha }\right]\quad\quad\;\;\;\end{equation} \tag{ 10 }$$

$$\begin{equation}{M_{{\text{TRM}}}} = {M_0} - {M_{\text{g}}}\,\exp \left[ - {\left(\frac{t}{\tau_p }\right)^\beta }\right].\end{equation} \tag{ 11 }$$

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In the above expressions, M₀ is the initial remanent magnetization, S stands for the magnetic viscosity, M_g denotes the glassy component of magnetization, whereas, τ and τ_p are the characteristic relaxation time constants. Here, the exponents α, β, and γ are constants out of which the former two are known as the Kohlrausch–Williams–Watt stretched exponents; usually these exponents are used to determine the strength of the dipolar interaction among the magnetic clusters [60–62]. The equations (8)–(10) all appear to fit well with the main panel data in figure 19, although, the best fit to the experimental data till the lowest measured time is obtained with only equation (10), which is confirmed from the logarithmic-time scale representation (inset of figure 19). However, our data does not fit to equation (11) for the expected range of β (0 ⩽ β ⩽ 1). Similar analysis was also performed for the M_TRM vs. t isotherms obtained at T = 6 K, 8 K, 9 K, 10 K, 12 K and 15 K, and the extracted parameters (along with the corresponding standard uncertainties) are listed in table 2.

Table 2. List of the scaling laws and corresponding fitted parameters obtained from the thermo-remanent magnetization measurements across T_SL.

Scaling laws	Fitting parameters	T = 5 K	T = 6 K	T = 8 K	T = 9 K	T = 10 K	T = 12 K	T = 15 K
MTRM = M0 − S Lnt	M0 (emu mol−1)	2310 ± 5	1871 ± 14	1802 ± 14	1962 ± 15	2141 ± 17	2283 ± 18	(58 ± 0.01) × 10−1
	S (emu mol−1)	46.4 ± 0.6	43.8 ± 1.7	32 ± 1.7	25 ± 1.9	19.3 ± 2.1	12.3 ± 2.2	(30 ± 0.01) × 10−2
MTRM = M0 ${t^{-\gamma }}$	γ	(218 ± 0.5) × 10−4	(259 ± 0.7) × 10−4	(192 ± 0.3) × 10−4	(134 ± 0.2) × 10−4	(93 ± 0.1) × 10−4	(55 ± 0.07) × 10−4	(662 ± 0.7) × 10−4
MTRM = M0 exp[–${\left( {\frac{t}{\tau }} \right)^\alpha }$]	τ (s)	(2.92 ± 0.13) × 108	(8.43 ± 0.35) × 107	(9.67 ± 0.66) × 108	(1.37 ± 0.1) × 1010	(2.4 ± 0.22) × 1011	(1.47 ± 0.17) × 1013	(11.5 ± 0.05) × 104
	α	(15.2 ± 0.06) × 10−2	(15.4 ± 0.06) × 10−2	(14.8 ± 0.08) × 10−2	(14.6 ± 0.07) × 10−2	(14.3 ± 0.08) × 10−2	(14 ± 0.08) × 10−2	(17.7 ± 0.02) × 10−2

The fitting parameters are plotted against temperature in figure 20 for further analysis. Evidently, the magnitudes of M₀ and τ falls drastically for T > 12 K while the exponents α and γ increase. It is noted that the values of all the fitted parameters obtained at 15 K are quite comparable to that observed in other SG systems like Nd₅Ge₃, PrRhSn₃, U₂PdSi₃ etc in the SG domain [63–65], probably because the system is in the SG state at 15 K. However, for T < 12 K when transition to SL state begins, we notice significant deviation both in the magnitudes and trend followed by these curves. In SG systems, with increase in temperature, α increases (τ decreases) monotonically throughout the SG domain and also beyond T_SG [63, 66, 67]. Similar pattern is exhibited by the ZCFMO system above ∼11.4 K, yet opposite trend is noticed below this temperature. According to the observations by Chu et al the slope change in the exponent may occur at a slightly higher temperature than where the change from the weak to strong irreversibility in magnetization takes place [67].

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The large magnitude of M₀ in the SL region is likely due to the presence of FM clusters or uncompensated spins [50]. Usually, for systems with a uniform energy barrier, α = 1 indicates that the system has a single relaxation time constant, whereas α = 0 corresponds to the absence of relaxation. For systems having a distribution of energy barriers, α ranges between 0 and 1. The value of α acquired in our case suggests that the ZCFMO system evolves through several intermediate metastable states, hence the activation occurs against multiple anisotropic energy barriers [35]. Additionally, contrary to SG systems, where increase in the value of the exponent (α or γ) with increasing temperature in the SG region reflects the increase of magnetic inter-cluster interaction, we observe in figure 20 weakening of interaction between the clusters with rise in temperature in the SL region [62]. Typically, in the SG ordered region, the magnetic viscosity S increases with increase in T as a signature of the thermally activated process. However, in ZCFMO, decrease in S with increasing T in figure 20 reveals that the properties in the SL region are not governed by thermal activation [68]. In support of the above results, the differentiation analysis ∂S/∂T vs. T (figure 20(b)) exhibits a sharp minimum across 9 K, as the ZCFMO system undergoes SL to SG magnetic phase transition.