Optimized Analysis of the AC Magnetic Susceptibility Data in Several Spin-Glass Systems using the Vogel-Fulcher and Power Laws

In spin-glasses (SG), the relaxation time $\tau$ ($= 1/2{\pi}f$) vs. $T_f$ data at the peak position $T_f$ in the temperature variation of the ac magnetic susceptibilities at different frequencies f is often fit to the Vogel-Fulcher Law (VFL): $\tau=\tau_0\exp[E_a/k_b(T_f-T_0)]$ and to the Power Law (PL): $\tau = \tau_0^*[(T_f-T_{SG}/T_{SG}]^{-z\nu}$. Both these laws have three fitting parameters each, leaving a degree of uncertainty since the magnitudes of the evaluated parameters $\tau_0$, $E_a/k_B$, $\tau_{0^*}$ and $z\nu$ depend strongly on the choice of $T_0$ and $T_{SG}$. Here we report an optimized procedure for the analysis of $\tau$ vs. $T_f$ data on several SG systems for which we could extract such data from published sources. In this optimized method, the data of $\tau$ vs. $T_f$ are fit by varying $T_0$ in the linear plots of $\ln \tau$ vs $1/ (T_f - T_0)$ for the VFL and by varying $T_{SG}$ in the linear plot of $\ln \tau$ vs. $\ln (T_f - T_{SG})/ T_{SG}$ for the PL till optimum fits are obtained. The analysis of the associated magnitudes of $\tau_0$, $E_a/k_B$, $\tau_{0^*}$ and $z\nu$ for these optimum values of $T_0$ and $T_{SG}$ shows that magnitudes of $\tau_{0^*}$, $\tau_0$ and $z\nu$ fail to provide a clear distinction between canonical and cluster SG. However, new results emerge showing $E_a/(k_BT_0)<1$ in canonical SG whereas $E_a/(k_BT_0)>1$ for cluster SG systems and the optimized $T_0<$ optimized $T_{SG}$ in all cases. Although some interpretation of these new results is presented, a more rigorous theoretical justification of the boundary near $E_a/(k_BT_0) \sim 1$ is desired along with testing of these criteria in other SG systems.


Introduction
The spin-glass (SG) phase has enamoured physicists since its discovery in the 1970s with Edwards and Anderson proposing the first basic model in 1975 [1].Giorgio Parisi winning the Nobel Prize in Physics in 2021 for "the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales" has made the SG problem as relevant as ever [2].Apart from the original problem of random magnets, it now serves to answer a widerange of questions catering to various fields of science from computer science to economy to biology [3][4][5][6].
The SG phase lacks long-range magnetic order, analogous to structural glasses which lack long-range structural order [7] and hence, the assigned SG nomenclature.The earlier studies on SG systems are summarized in the review by Binder and Young [8] and in the book by Mydosh [9].The two well-known classes of SG are: (i) canonical SG consisting of individual magnetic impurities present in a non-magnetic system with long-range inter-spin interactions such as AuMn containing 2.98% Mn; and (ii) cluster SG usually consisting of magnetic ions with shortrange inter-spin interactions and larger concentrations of magnetic ions but below the percolation threshold.Generally, the SG phase is characterised by multiple equilibrium ground states like free-energy valleys separated by energy barriers.Hence the SG phase constitutes a new state of magnetism different from the long-range ordered (LRO) magnetic states like ferromagnets, ferrimagnets, and antiferromagnets.However, in SG systems, the collective freezing of spins does occur below a freezing temperature TSG but without LRO, typically resulting from random site occupancy and geometric frustration along with competing exchange interactions.
Experimentally, the SG phase shows some distinct features: (i) a frequency-dependent shift of the peak temperature Tf in the temperature dependence of the ac magnetic susceptibility (χ' and χ'') data; (ii) absence of an anomaly or at best a broad maximum around TSG in the specific heat Cp vs T data instead of a lambda type asymmetric peak observed for LRO systems; (iii) non-exponential time dependence of magnetization, and (iv) memory effects [8,9].The spin relaxation in SG systems is due to the multivalley ground state, these valleys being separated by temperature-dependent energy barriers of different magnitudes [8,9].On cooling the system through TSG, the system lands in one of these valleys from which it can relax by overcoming these barriers.This makes the relaxation in SG dependent on external perturbations such as applied magnetic field and ac frequency.The bifurcation of the field-cooled and zero-field cooled magnetization below TSG, shift of TSG with applied H given by the de Almeida -Thouless [10] or Gabay-Toulouse [11] lines, non-linear susceptibility, magnetic viscosity, aging phenomenon, and the frequency-dependent shift of the peak temperature Tf in the ac susceptibility are some of the techniques used for investigating spin relaxation in SG systems [8,9,10,11].
In this work, we have focussed on the analysis of Tf vs. f data obtained from ac susceptibility studies in seventeen (17) SG systems for which we could extract the data from published sources.Temperature and frequency dependence of the ac susceptibilities, χ' and χ'', are often used to distinguish between magnetic nanoparticles and canonical vs. cluster SG systems.For this purpose, the temperature shift in the peak position Tf in χ'' or χ' with change in frequency f is used to define the Mydosh parameter Ω [8,9,12,13]: where  1 and  2 are two sufficiently different frequencies.For magnetic nanoparticles, it is generally observed that Ω > 0.05 with the magnitude of Ω increasing with decrease in the interparticle interaction [13].For 0.01< Ω < 0.05, the system is usually classified as a cluster SG, whereas for a canonical SG system, Ω ~ 0.005 is an order of magnitude smaller [8,9,12,13].
Additional information on the spin dynamics of such systems is obtained from the fit of the relaxation time τ vs. Tf data to the Vogel-Fulcher law (VFL) given by [8,9,16]: and the Power Law (PL) given by [8,9,[17][18][19][20]: For use in Eqs. ( 2) and ( 3), the relaxation time τ =1/2πf is determined from the frequency f whereas Tf represents the temperature of the peak positions in the ac susceptibilities χ' (or χ'') vs. T data at different f thus yielding the τ vs. Tf data.Other parameters in Eqs. ( 2) and (3) are: activation energy Ea, zυ,  0 , and τ 0 * , the relaxation times of individual spins or clusters; T0, the strength of interparticle or inter-cluster interactions;   , the SG temperature; and zυ, the dynamical critical exponent.In general, for most reported SG systems, zν lies between 4 and 12 and the magnitude of τ0 * usually falls between 10 −12 and 10 −13 s for canonical SG systems, whereas, in general, τ0 * for the cluster SG is significantly higher and lies in the range 10 −7 to 10 −10 s [8,9,19,[21][22][23][24][25][26].However, the consensus about the range of values for these parameters is often contradictory [27][28][29][30] and how the magnitudes of TSG and T0 are selected is often not explained.
The determination of Ω from the data using Eq. ( 1) is straightforward.However, fitting the τ vs. Tf data to Eqs. ( 2) and ( 3) is possible for a range of these fitting parameters because both these laws have three fitting parameters each, thus leaving a degree of uncertainty.Specifically, the magnitudes of the evaluated parameters τ0, Ea/kB, τ0 * and zυ depend on the choice of T0 and TSG.As noted in our recent paper on the spinel ZnTiCoO4 (listed as ZTCO hereafter) [31], the linear plots of Ln τ vs 1/(Tf -To) for different choices of T0 for the VFL and Ln τ vs. Ln (Tf -TSG)/TSG for different choices of TSG for the PL are possible.These linear fits yield τ0 and Ea/kB for each choice of T0 for the VFL and τ0 * and zν for each choice of TSG.Using different values of TSG and T0 still yielded respectable linear fits to Eqs. ( 2) and ( 3), yet magnitudes of the fitting parameters were quite different for each choice of TSG and T0.So, a procedure was developed by plotting the variations of the evaluated parameters and associated adjusted R 2 (AR 2 ) value of the linear fits against T0 and TSG [32].The quality of the linear fits was determined from the maximum value of AR 2 , with AR 2 =1 being valid for a perfect linear fit [32].The analysis of the data for ZTCO yielded the optimum (maximum) value of AR 2 = 0.993 for a particular   and T0 [31].The corresponding magnitudes of  0 and   in Eq. ( 2) and τ0* and zυ in Eq. (3) were then considered as optimum magnitudes of these parameters for the system.
In this paper, we have employed this procedure to reanalyse the data of τ vs. Tf in seventeen (17) randomly chosen SG systems for which we could extract such data from published sources.Results of the evaluated parameters τ 0 and   /  in Eq. ( 2) and τ0* and zυ in Eq. (3) from this optimized analysis for the seventeen SG systems along with the evaluated magnitudes of Ω are collected in Table I with details of the analysis given in the following pages.This analysis shows that magnitudes of τ0* and zυ alone are inadequate to distinguish between different classes of SG.Instead, it is proposed that the magnitudes of Ω along with that of the ratio Ea/(kBTo) can be used to distinguish between the canonical and cluster SG unambiguously.Typically, Ω ∼ 10 -3 for canonical SG while Ω ∼ 10 -2 in cluster SG as also previously reported in literature and used by many investigators to distinguish between canonical and cluster SG as also discussed below.The new results reported below from the analysis presented in this work are that the ratio Ea/kBT0 < 1 in canonical SG whereas Ea/kBT0 >1 for cluster SG, and the optimized T0 < optimized TSG.Some interpretations of these new results are also presented along with suggestions for future studies.

Examples of various spin-glasses
Analysis of the  vs. Tf data in seventeen (17) SG systems is presented here using the peak temperature Tf of the ac susceptibilities and τ = 1/2πf for each frequency f.For these systems, we could extract the data of Tf vs f from the ac susceptibilities from published sources using the software, WebPlotDigitizer-4 [33].But for reasons of brevity, we show the details of the data and analysis for only two systems whereas corresponding figures for the other 15 systems are given in the SI (supplemental Information [34]).Typically, these measurements were done with ac magnetic field with amplitude ~ 5 Oe.
The results in Fig. 2 show that with increase in TSG, magnitude of τ0* increases rapidly whereas zν decreases almost linearly.Similarly, with increase in T0, τ0 increases rapidly whereas Ea/kB decreases linearly.This is a common theme for all the results presented here, necessitating the need for determining the optimum values of TSG and T0 as reported in this work.

CuMn:
The data of Tf vs. f for the CuMn4.6%system was reported in 1980 by Tholence whose initial analysis of this data assuming τ0 =10 -13 s for the VFL yielded T0 =25.5K and Ea/kB = 59 K [36].However, a more detailed analysis by Souletie and Tholence in 1985 [37] reported the following fitting parameters for this system: TSG =27.45 K with τ0 * = 7.7×10 -13 s, and zν = 5.5 for the PL and T0 = 26.9K with τ0 = 4.0×10 -8 s and Ea/kB =11.75 K for the VFL.Our analysis of this  , zυ and TSG in the Power Law and optimum magnitudes of T0, τ0 and Ea/kB in the Vogel-Fulcher Law.See Table I for listing of the optimum parameters for different systems.
data with maximum AR 2 = 0.9975 (shown in Fig. S1) yielded the following optimum parameters: s, and zν = 6.82 for the PL and T0 = 26.7 K with τ0 = 8.8×10 -10 s and Ea/kB =16.7 K for the VFL.These optimum values from our analysis, also listed in Table I, are only slightly different from those reported in [37].With Ω = 0.0084 and Ea/kBT0 = 16.7/26.7= 0.625 < 1, this system is classified as a canonical SG.

Na0.7MnO2: Luo et al reported experimental studies on the hexagonal SG candidate
Na0.70MnO2 including the frequency-dependent ac magnetic susceptibility measurements [38] with TSG = 39.0K and Ω = 0.004, the latter falling in the range observed in canonical SG.By extracting the data of χ' vs. T at different f from [38], we carried out our analysis like the one shown in Fig. 2 for AuMn, see Fig. S2 in SI [34].Results from our analysis yielded Ω = 0.005, optimum TSG = 38.6K, τ0* = 1.5×10 -14 s, and zν = 6.0, optimum T0 = 38.1 K, τ0 = 1.3×10 -8 s, and Ea/kB =12.3 K, the latter yielding Ea/kBT0 = 0.32 < 1.Using our criteria, Na0.70MnO2 is a canonical SG.This agrees with the conclusion of [38] which was based just on the magnitude of Ω = 0.004 since analysis of the data using the PL and VFL was not reported.

Zn3V3O8:
Reporting on the bulk magnetic properties of vanadium-based geometrically frustrated system Zn3V3O8, Chakrabarty et al noted that the SG state in Zn3V3O8 originated from clusters of atoms rather than individual atoms and so identified the system as a cluster glass [42].
Here we present results obtained from similar analysis of the  vs. Tf data obtained from the peaks in χ' and compare the values obtained in the two cases.This analysis, shown in Fig. S8 [34], yielded Ω = 0.033, TSG = 13.2K, τ0* = 4.3×10 -11 s, zν = 11.1, optimum T0 = 10.9K, τ0 = 2.1×10 -12 s and Ea/(kBT0) = 9.44 > 1.Although the optimum T0 is identical in the two cases, magnitudes of the optimum TSG and all other parameters are slightly larger for the χ' data.This may be related to the fact that Tf for χ' occurs at a slightly higher temperature than that for χ'', and Tf for χ'' coincides with peak in d(χ'T)/dT [43].However, both data sets yielded similar trends in their values and indicate towards the same conclusion that ZTCO is a cluster SG. 2.2.5.Co2RuO4: Ghosh et al reported magnetic investigations of the spinel Co2RuO4, which they classified as a cluster SG [21].They analyzed the data of χ'' vs T by employing the standard scaling laws and reported Ω = 0.01, TSG = 14.97 K, τ0* = 1.16×10 -10 s, zν = 5.2, T0 = 14.3K and τ0 = 1.1×10 - 9 s.Our results from the new analysis are shown in Fig. S9 [34] yielding Ω = 0.018, optimum TSG = 14.5 K, with τ0* = 2.6×10 -12 s, and zν = 8.5 and optimum T0 = 13.5 K, τ0 = 6.2×10 -10 s and Ea/(kBT0) = 2.55 >1.Using the criteria of Ea/(kBT0) >1, Co2RuO4 is a cluster SG, in agreement with the conclusion in [21] although the magnitudes of the optimum parameters determined here are different from those reported in [21].

Interpretation
To explain the result of TSG > T0 evident from our optimized parameters listed in Table 1, Eqs. ( 2) and ( 3) are rewritten as: Ln(τ/τo) = Ea/kB(Tf -T0) ---( 4) Following Souletie and Tholence [37], we use Ln(τ/τo) ~ 20 and Ln(τ/τo*) ~ 25.Eqs. ( 4) and (5) can then be solved yielding From Eq. ( 6), (TSG -T0) can be calculated using optimized values of Ea/kB, TSG and zν listed in Table 1 for each case and compared with the optimized fitted value of (TSG -T0).This comparison of the "experimental" and "calculated" values of (TSG -T0) for all the systems discussed here are also given in Table I, showing that the calculated (TSG -T0) is indeed positive, meaning TSG > T0 although the agreement between the "experimental" and "calculated" values of (TSG -T0) is better in some cases than in other cases.This difference is likely related to some difference in the approximate values of Ln(τ/τo) ~ 20 and Ln(τ/τo*) ~ 25 used here for all cases in the calculations.
This explanation of the result TSG > T0 provides additional confidence in the correctness of the optimized analysis presented in this paper.
The optimized analysis in seventeen SG systems presented here shows that canonical and cluster SG systems have different magnitudes of the ratio Ea/(kBT0) and the Mydosh parameter Ω in that Ea/(kBT0) <1 along with Ω < 0.01 for canonical SG, and Ea/(kBT0) >1 along with Ω > 0.01 for cluster SG.In contrast, the associated magnitudes of τ0*, τ0, and zν are found to be less reliable in distinguishing between different SG system since no systematic pattern from Table I is evident in their variations between canonical and cluster SG.All SG systems consist of spin clusters [8,9], the difference between canonical and cluster SG is the size of the clusters and energy barrier between the ground state valleys of various clusters.The condition Ea/(kBT0) <1 implies that in canonical SG, the energy barrier Ea/kB is smaller than the inter cluster coupling T0 whereas for Ea/(kBT0 )> 1 valid for cluster SG, Ea/kB > T0.

Concluding Remarks:
In this paper, we have presented results using a new optimized procedure for fitting the data of τ vs. Tf to the VFL (Eq.2) and the PL (Eq. 3) for seventeen SG systems for which we could extract such data from published sources.This analysis showed how the parameters τ0 and Ea/kB for the VFL and τ0 * and zυ for the PL strongly depend on the choice of T0 and TSG respectively.
Hence, to eliminate this uncertainty, we employed the maximum AR 2 method to determine the optimum values of T0 and TSG, and it is suggested that the magnitudes τ0, Ea/kB, τ0 * and zυ associated with these optimum values of TSG and T0 should be used in the discussion, analysis, and classification of SG systems.Results from this analysis show that magnitudes of τ0 * , τ0 and zυ fail to provide a clear distinction between canonical and cluster SG.However, new results emerge showing Ea/(kBT0) < 1 in canonical SG systems, and Ea/(kBT0) >1 for cluster SG systems.Also, the optimized TSG > optimized T0 in all cases.More rigorous theoretical interpretation of the boundary near Ea/(kBT0) ~ 1 is desired along with testing of this criterion for distinguishing between canonical and cluster SG systems as appropriate data in such systems become available in the future.

Supplementary Material:
The Supplementary material contains graphs representing the optimized procedure for the analysis of τ vs. Tf data in fifteen SG systems, excluding that for AuMn2.98% and PrRhSn3, analysed in this work.The Supplementary Material has been assigned the reference number [34] for use in the main text.

Please Note:
The following article has been submitted to AIP Advances.

1 /
(Tf -T0) for the VFL and Ln τ vs. Ln (Tf -TSG)/TSG for the PL for different choices of T0 and TSG were made with the linear fits yielding τ0 and Ea/kB for each choice of T0 in case of the VFL and τ0* and zν for each choice of TSG for the PL.These plots depicting the variations of the evaluated parameters for AuMn with change in T0 and TSG are shown in Fig 2.Here, the optimum TSG = 10.0K with maximum AR 2 = 0.997 is observed for the PL analysis with the corresponding optimum τ0 * = 3.6×10 -16 s, and zν = 7.6 whereas in the VFL analysis, peak with AR 2 = 0.997 occurs at T0 = 9.7 K with τ0 = 1.7×10 -11 s and Ea/kBT0 = 0.98.Our analysis also yielded Ω = 0.006 for AuMn.According to our suggested classification (Ea/kBT0 < 1 for canonical SG and Ea/kBT0 >1 for cluster SG), AuMn is identified as a canonical SG.

Figure 1 :Figure 2 of
Figure 1: Frequency dependence of the real part of ac susceptibility χ' versus temperature extracted from Figure 2 of Mulder et al[35] around the freezing temperature of AuMn2.98% using the software WebPlotDigitizer-4[33].

Figure 2 :
Figure 2: Plotted are the variations of the evaluated parameters for AuMn2.98% as a function of different choices of TSG in the Power Law (Eq. 3) and as a function of T0 in the Vogel-Fulcher Law (Eq.2) with the dotted lines connecting the data points as visual guides.The vertical dotted lines mark the positions of the maximum AR 2 and the corresponding optimum magnitudes of  0 *

Figure 3 :
Figure 3: Temperature dependence of the ac-magnetic susceptibility χ'' measured at different frequencies extracted from Figure 2(b) of Anand et al[19] around the spin-freezing temperature of PrRhSn3 using the software WebPlotDigitizer-4[33].The black arrows mark the peak positions for the χ'' versus T data at different frequencies.

Figure 4 :
Figure 4: Same as Fig. 2 except the plots and analysis are for PrRhSn3 using the information from Fig. 3.
scheme.M.R.C. and S.T. acknowledge the FIST program of the Department of Science and Technology, India for partial support of this work (Grants No. SR/FST/PSII-020/2009 and No. SR/FST/PSII-037/2016).M.R.C. and S.T. acknowledge the financial support from UGC-DAE CSR through a Collaborative Research Scheme (CRS) Project Number CRS/2021-22/01/383.S.T. acknowledges the DST SERB Core Research Grant File No. CRG/2022/006155 for the partial support of this work.M.R.C. and S.T. acknowledge the Central Instrument Facility of the Indian Institute of Technology, Guwahati for support.M.R.C. expresses her sincere thanks to Dr. Sayandeep Ghosh for useful discussions.

Figure S2 :
Figure S2: Same as for Fig. S1 except the plots and analysis are for Na0.7MnO2.

Figure S3 :Figure S4 :
Figure S3: Same as for Fig. S1 except the plots and analysis are for CaSrFeRuO6.

Figure S10 :Figure S11 :
Figure S10: Same as for Fig. S1 except the plots and analysis are for Pr4Ni3O8.

Figure S12 :
Figure S12: Same as for Fig. S1 except the plots and analysis are for Zn0.5Ni0.5Fe2O4.

Figure S13 : 2 Figure S14 :
Figure S13: Same as for Fig. S1 except the plots and analysis are for LiFeSnO4-LT.

Figure S15 :
Figure S15: Same as for Fig. S1 except the plots and analysis are for ScDy3.5%.