Introduction

Spin glasses1,2 display dynamical properties such as aging, memory, and rejuvenation3. Aging reflects a slow equilibration of the spin configuration at a constant temperature in the spin glass phase after a quench from high temperature. Such an aged spin configuration is kept in memory upon further cooling while at the same time new spin configurations are imprinted at shorter lengths scales (rejuvenation)3,4. To probe the intrinsic response of the spin glass phase, magnetometry experiments are performed in low magnetic fields, so that a linear response to field changes is achieved and the principle of superposition applies to the magnetic relaxation4. Interacting magnetic nanoparticles have been found to display glassy dynamics and undergo (super)spin glass phase transitions, e.g. concentrated ferrofluids5,6, dense assemblies of nanoparticles7,8,9 and nanocomposites10,11. The magnetic properties of magnetic nanoparticles are affected by the nature and strength of the inter-particle magnetic interaction11, magnetic anisotropy12, the nanoparticle size and shape distribution9, as well as the interplay between inter- and intraparticle interactions stemming from surface13,14.

Glassy magnetic states have been reported in MnFe2O415 nanoparticle systems of various sizes16,17. In the current study, we have investigated the superspin glass state of a dense assembly (i.e. particles are in close contact) of ultra-small (diameter 2 nm) MnFe2O4 nanoparticles. This implies a significant role of the disordered surface affecting both intra (core/surface exchange coupling) and interparticle (dipolar and exchange interactions) effects. We observe that the system undergoes a superspin glass phase transition and we provide evidence, by means of time and temperature dependent memory experiments, that the low field magnetization dynamics is similar to that of conventional (atomic) spin glasses. Monte Carlo simulations, using a mesoscopic model of the nanoparticle system, reproduce the experimental findings, and confirm, accounting for core-surface and interparticle contributions, the significance of the interplay between intra- and interparticle effects. The relative contribution of dipolar and exchange (involving the surface shells) interparticle interactions is determined by simulating the memory experiment in absence of one of them. The simulations indicate that both types of interactions contribute to the observed memory effect and that the contribution of dipolar interactions is much stronger than that of exchange ones, which alone would produce a much smaller effect.

Results and discussion

Earlier studies on the MnFe2O4 powder18,19 showed that the mean crystallite size obtained by XRD analysis is ~ 2 nm, whereas the value obtained by powder specific area (278 m2/g) is about 4 nm. This discrepancy suggests that the particles consist of a few aggregated crystallites. Remanence magnetization measurements by means of DCD and IRM protocols recorded at 5 K clearly show the presence of long range dipolar interparticle interactions. Detailed discussion of these results is reported in the Supplementary Materials.

To further investigate the magnetic structure of these small particles, 57Fe Mössbauer spectrometry under intense external magnetic field (MSMF) was performed; all the spectra were analyzed using the program Mosfit20. MSMF allows a more reliable distinction between Fe ions located in interstitial sites with tetrahedral (A) and octahedral (B) oxygen coordination (the applied field is usually added to the A-site hyperfine field and subtracted from the B-site hyperfine field allowing a smaller overlap between the two components than in zero field Mössbauer spectra)21,22. Furthermore, MSMF spectra can also give information about the magnetic structure of the nanoparticles. In the presence of an external magnetic field parallel to the gamma ray direction, the relative areas of the six lines give information about the degree of alignment of the magnetic moments with the applied field.

The 57Fe Mössbauer spectrum (Fig. S2) has a complex shape and it is broadened while the intermediate lines show somehow high intensity and the broadening of the B-sites lines is more pronounced than the A-sites lines. A three-component model was necessary to fit the spectrum (see Supplemental Materials for details). MSMF spectrum indicates the presence of a ferrimagnetic (FiM) and an antiferromagnetic (AF) like phase. This result is in agreement with the Rietveld analysis, identifying the two phase as MnFe2O4 and (Mn1/3Fe2/3)O2/3(OH)4/3, probably arranged in a core–shell structure19. MnFe2O4 has ferrimagnetic structure, whilst a high frustration due to a reduced symmetry23 induces an AF-like behavior.

In order to investigate magnetization dynamics of the nanoparticles, AC susceptibility and DC magnetization measurements and zero field Mössbauer spectrometry at different temperature have been performed. Figure 1a shows ZFC /FC/ TRM magnetization curves (see “Methods” and Supplemental Materials for a description of these experiments). The FC and ZFC curves coalesce at temperatures just above the maximum in the ZFC, indicating a superparamagnetic behavior of the nanoparticles at higher temperatures. This is confirmed by M Vs H curves recorded between 100 and 200 K which collapse on a single curve when plotted as M Vs H/T (inset Fig. 1a), confirming the superparamagnetic behavior of the nano-entities at these high temperatures24,25. The superparamagnetic onset above Tmax is also confirmed by thermoremanent magnetization, which approaches zero at the temperature where FC and ZFC curves merge. MZFC shows a maximum at Tmax ~ 45 K; below this temperature MFC shows a maximum (~ 43 K), then becomes temperature independent and finally shows a slight upturn at the lowest temperature. This behavior resembles that of spin glasses3 and has been observed in monodisperse systems of strongly interacting magnetic particles7,8,9. As discussed in Supplementary Materials (Sect. 5), the presence of strong magnetic interaction between the particles is confirmed by magnetic-field dependent remanent magnetization measurements by means of DCD and IRM protocols recorded at 5 K.

Figure 1
figure 1

(a) ZFC (full circles), FC (empty circles) and TRM (empty squares) curves recorded in H = 10 Oe; inset M Vs (H/T) curves recorded in the range 100 – 200 K; (b) 57Fe Mössbauer spectra of MnFe2O4 samples recorded in the range 4–65 K.

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Figure 1b shows 57Fe Mössbauer spectra recorded at different temperatures in the range 4–65 K. Mössbauer spectra of magnetic nanoparticle assemblies typically consist of a superposition of a sextet due to particles with long relaxation time compared to the time scale (⁓ 5 × 10−9 s) of Mössbauer spectrometry and a doublet due to particles with shorter relaxation time compared to it. The relative area of the doublet increases with increasing temperature. The blocking/freezing temperature in Mössbauer spectroscopy TMoss, may be defined as the temperature at which the spectral areas of the sextet and the doublet are equal. From analyses of our Mössbauer spectra at different temperatures (some spectra are shown in Fig. 1b), TMoss ≈ 62 K has been determined. Mössbauer spectroscopy and DC magnetization measurements have significantly different time scales (⁓ 10–30 s for magnetization measurements), and thus the freezing/blocking temperature Tmax estimated using the two techniques are expected to differ considerably. In the literature, it is reported that the ratio TMoss/Tmax decreases with increasing interparticle interactions26,27. For the blocking temperature of non-interacting particles, obeying Arrhenius dynamics, the ratio TMoss/Tmax is about 6, whilst lower values are reported for spin glass and superspin glass systems28, where critical dynamics govern the behavior. For our sample the ratio TMoss/Tmax is ⁓ 1.4.

The ac-susceptibility was recorded for different frequencies (f) and the in-phase component of the susceptibility χ’(T,f) is shown in Fig. 2a. Considering the FC magnetization data (plotted as M/H) as equilibrium susceptibility, one can see that the various χ’(T,f) curves closely follow the equilibrium curve above 50 K in the superparamagnetic state. Below this temperature, the curves start to deviate. We can define a freezing temperature Tf for each frequency, below which the longest relaxation time (τ) of the system exceeds the observation time of the measurement, τ ~ 1/(2πf). (See Fig. S3 in Supplementary Materials). Such a freezing temperature may be defined for each frequency f, yielding (Tf ,f) datasets which may be analyzed using various scaling laws. Data obtained from Mössbauer spectroscopy experiments (τ = 5 × 10−9 s, Tf = 62 K) is added using an open marker. In spin glass systems, the critical slowing down implies that the time necessary to reach equilibrium will become longer and longer when approaching the spin glass phase transition Tg (glass temperature) and τ diverges at Tg according to the power law τ/τ0 = ε−zν, where ε is the reduced temperature (T − Tg)/Tg and z, ν critical exponents2. As seen in Fig. 2b, the Tf data follows a power law behavior on a pretty wide frequency window, with physical values of critical exponents (zν = 9(1)) and flipping times, (τ0 = 10−12 (1) s) indicating a (super)spin glass transition at Tg = 46 (1) K in the system. The error bars on the parameters derived from the scaling analysis are quite large. Yet, the obtained zν value is comparable to those of 9–11 reported for superspin glasses5,9, and in the range of those observed for "isotropic" Heisenberg atomic spin glasses (zν ~ 6–8) and "anisotropic" Ising ones (zν ~ 10–12)29. The value of τo determined in our analysis is lower than typical values for superspin glasses5, possibly due the very small size and strong magnetic interaction of the particles.

Figure 2
figure 2

(a) Temperature dependence of the in-phase component of the ac-susceptibility recorded for different frequencies; field amplitude h = 4 Oe (f = 1.7, 5.1, 17, 55, 170 Hz), and h = 17 Oe (f = 170, 510, 1700, 5100 Hz). The temperature dependence of the field cooled (FC) magnetization collected in H = 10 Oe is added for comparison. All data is normalized by χ' (f = 1.7 Hz, T = 60 K). (b) scaling of the relaxation time τ with the reduced temperature ε = (T − Tg)/Tg. Filled markers represent the (Tf) data obtained from the ac-susceptibility data; open marker is obtained from Mössbauer spectrometry.

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The existence of non-equilibrium dynamics of MnFe2O4 is investigated by means of time-dependent relaxation (not shown) and temperature-dependent memory experiments performed in small magnetic fields (10 Oe), described in the Supplemental Materials (see Fig. S5 for a sketch of the measurement protocols). Figure 3a,b show the ZFC and TRM reference curves (presented in Fig. 1a), as well as the corresponding curves recorded on reheating after a halt of duration th = 10800 s at Th = 20 K without magnetic field change (memory curves). In the TRM case the field is thus kept to its H = 10 Oe value, while for the ZFC, it remains zero. The memory curves show a dip (ZFC) or bump (TRM) illustrating the memory and rejuvenation effects. The FC magnetization experiences a minute downward relaxation during such a halt in a constant magnetic field. The principle of superposition connects the response to magnetic field changes and relaxation as MZFC(tw,t) = MFC(0,t+tw) − MTRM(tw,t)3, provided that the field change yields linear response. Using a specific heating rate in ZFC/TRM experiments corresponds to probing the system at a specific observation time (order of 10 s). The similarity of the difference plots between reference and ZFC/TRM memory curves plotted in Fig. 3c hence indicates that the principle of superposition observed in the spin glasses3 is valid also for our sample. The excess magnetization at Th in MTRM (T) is equal to the magnetization loss in MZFC (T) in absolute values.

Figure 3
figure 3

(a) ZFC and (b) TRM magnetizations for H = 100 Oe. Two curves ZFC and TRM measured after a th = 10800 s halt at Th = 20 K during cooling are added; (c) difference with the corresponding reference curves.

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Memory experiments have been also performed using MC-simulations, considering a mesoscopic model for a dense assembly of ferrimagnetic nanoparticles showing superspin glass characteristics in ZFC-FC curves, as described in Ref13. The energy parameters given in the Supplementary Material (equation S1) are based on the bulk values of MnFe2O4 (MS = 5 × 105 A/m and K = 3 × 103 J/m3), and their modifications are established considering the nanoparticles morphology (e.g. reduced symmetry and reduced size) using a mean field approach. The three spin model was employed and the effective intra-particle exchange coupling constants among the core spin and the surface spins were taken as jc1 = 0.5, jc2 = 0.45, jsrf = − 1.0 and the effective anisotropy constants of the core as kC = 0.05 and the surface ksrf = 1.0. We take the inter-particle exchange coupling constant as jinter = − 0.50 as a free parameter, the dipolar strength is calculated and found g = 3. Figure 4 shows the Monte Carlo simulations for the reference ZFC and TRM magnetization curves together with the memory curves for tw = 5 × 106 and their difference ΔΜ. The dip of the memory ZFC, the bump of the memory TRM and the similarity of the ΔΜ curves of the two memory experiments confirms that the dynamic properties of the simulated system are characterized by the existence of a superspin glass state. The validity of the model, accounting for interparticle contributions and intraparticle (i.e., intraparticle exchange interaction, namely core/surface and surface exchange interaction, and surface anisotropy), demonstrates that the observed memory effects result from an interplay between interparticle interactions and surface disorder.

Figure 4
figure 4

(a) Monte Carlo simulations of the ZFC (a) and TRM (b) magnetizations at H=0.08. Two curves ZFC and TRM calculated after a tw = 5*106 MCSS stop at T = 0.18 while cooling are added; (c) difference with the corresponding reference curves.

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In order to determine the relative contribution of dipolar and exchange (involving the surface shells) interparticle interactions, we have simulated the memory experiment in absence of one of them, i.e. switching off either jinter or g, respectively. The results (Fig. 5) indicate that both kinds of interactions contribute to the observed memory effect. It is also evident that the contribution of dipolar interactions is much stronger than that of exchange interactions, which alone would produce a much smaller effect (Fig. 5b; absence of dipolar interactions), as we have also pointed out in reference13 (see also Supplemental Materials for some insight on intra-particle interaction effects). Weak rejuvenation effects have been reported in many cases for atomic spin glasses in simulations based on the Edwards Andersson model30. Similar results have been obtained for Ising models of dipolarly interacting nanoparticles31. However the present Heisenberg model takes into account the interparticle exchange interactions and the anisotropy energies of the weakly anisotropic Mn ferrite nanoparticles together with strong surface spin disorder, and hence may display stronger memory and rejuvenation features.

Figure 5
figure 5

Difference ΔΜ between ZFC reference and ZFC memory curves for: (a) the system in absence of dipolar interactions (g = 0); (b) the system in the absence of exchange interparticle interactions (jinter = 0).

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The memory experiments described above corresponds to field stop (FS) experiments in which the magnetic field is kept to its initial value (zero for ZFC and H for TRM) during the time th. The influence of aging (wait time tw dependence) on the magnetic relaxation can be directly monitored in low field isothermal remanent magnetization vs. temperature experiments (MIRM(T)) using a similar temperature protocol as in the memory experiments including a field application after different wait times (tw) during the halt3,32. (tw is the time the sample has been kept at constant temperature before the field change). The excess magnetization attained during the halt freezes in when the field is cut off and cooling resumes after the halt. The inset of Fig. 6a shows MIRM(T) recorded on heating using tw = 0 and tw = 3000 s and in both cases a hold time for the magnetic field of 3000 s. As seen in the figure, the magnitude of MIRM depends on the wait time at Th (= 25 K) before the field application, which reflects a wait time dependence (aging) of the zero field cooled magnetization MZFC(t,tw). Similar experiments may be devised, considering a zero-field stop (ZFS) procedure for the TRM, in which the magnetic field is switched to zero during the halt3. Sketches of the evolution of the temperature and magnetic field with time in the various protocols are included in supplementary materials (Fig. S5). The results of the ZFS and FS MTRM(T,tw) measurements using the same wait times and hold times as in the IRM experiments are shown in Fig. 6a (inset).

Figure 6
figure 6

(a) TRM magnetization measured after Field Stop (FS) with 10 Oe and Zero Field Stop (ZFS) at 25 K after different waiting (tw) and halt time (th) as defined in the text ; (inset a) IRM magnetization measured after FS with 10 Oe at 25 K for different tw and th as defined in the text; (b) different between pair of curves reported in (a): ΔMIRM, ΔMTRMFS (plotted as − ΔMTRMFS ) and ΔMTRMZFS (plotted as − ΔMTRM ZFS) for MnFe2O4 (main frame) and the archetypal spin glass Ag(Mn) (inset).

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Both the IRM and TRM data (ZFS and FS conditions), reflect, as expected, the aging phenomenon. Comparing the curves collected without wait time (tw = 0) or including a wait time (tw = 3000 s) at the halt temperature before the field application, one may observe how the IRM curves recorded with tw = 3000 s lie significantly lower than the tw = 0 one3,32. The difference curves between the experiments performed with and without wait times, denoted ΔMTRMZFS, ΔMTRMFS and ΔMIRM are plotted in Fig. 6b. It was argued in Ref.3 that ΔMTRMZFS(T), ΔMIRM(T), and ΔMTRMFS(T) reflect MTRM, MZFC, and MFC respectively, at an observation time given by the heating rate (the same in all these experiments). The superposition of relaxations MZFC(tw,t) = MFC(0,t+tw) − MTRM(tw,t) would then imply the relation ΔMIRM(T) = ΔMTRMFS(T) − ΔMTRMZFS(T). This seems to be satisfied for the excess magnetization of the MnFe2O4 superspin glass, as we observe that ΔMTRMZFS(T) (plotted as − ΔMTRMZFS(T)) is quite similar to ΔMIRM(T), except in the vicinity of Th, around which ΔMTRMFS(T) is non-zero. As seen in Fig. 6b, the ΔMTRMZFS(T), ΔMIRM(T), and ΔMTRMFS(T) curves for MnFe2O4 (main frame) are quite similar to those of the archetypal Ag(Mn) spin glass (inset)3.

The IRM magnetization curves presented in Fig. 6a for MnFe2O4 are relatively flat at low temperatures, and decrease above Th. This behavior is qualitatively different from the above mentioned Ag(Mn) spin glass, for which an upturn of the magnetization is observed below Th32,33. In the latter case, the upturn has been related to the spin dimensionality (Heisenberg-like), as Ising spin glasses show IRM curves without upturn, akin to MnFe2O432. Interestingly, it was observed in Ref.33 that the IRM curves of maghemite nanoparticles had different superspin dimensionality, depending on Th. The “Heisenberg character” of the IRM curves at temperatures near Tg (with upturn below Th) is gradually replaced by an “Ising character” as Th becomes lower and lower. Interestingly a similar crossover is observed in the present MnFe2O4 system (see Fig. 7) reinforcing, as predicted in Ref.33, the idea of an influence of the individual particle relaxation and anisotropy on the apparent superspin dimensionality of the system.

Figure 7
figure 7

MIRM vs temperature using a perturbation field of 50 Oe and th = 300 s for several halting temperatures; Th = 20, 25, 28, 31, 34 K. The data was corrected for a background contribution, as illustrated in the inset. The inset shows the magnetization data without correction at Th = 25 K and 31 K as well as the reference background measurement, denoted IRM ref, in which the system is cooled and measured in zero applied magnetic field.

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Conclusions

The dynamical magnetic properties of a superspin glass consisting of a dense assembly of ultra-small MnFe2O4 nanoparticles have been investigated by means of SQUID magnetometry. The zero-field cooled (ZFC), thermoremanent (TRM), and isothermal remanent magnetization (IRM) were recorded, using specific protocols, to compare the low field magnetization dynamics of that system to that of archetypal spin glasses. Mössbauer spectra were recorded as a function of temperature and magnetic field, yielding information on the dynamical magnetic properties and magnetic structure, respectively. The low-field magnetization dynamics was found similar to that of archetypal spin glasses in spite of the inherent characteristics of nanosystems (superspins instead of atomic spins, surface disorder, interparticle and intraparticle magnetic interaction), reflected in the observed superspin dimensionality crossover. The time and temperature dependence of the ZFC and TRM magnetization were well reproduced by Monte Carlo simulations, using a mesoscopic model of a nanoparticle assembly, with core/shell morphology, accounting for all the intraparticle (core, surface, core/surface interface coupling) and interparticle (dipolar and exchange interactions) effects. This provides an evidence of the interplay between the above effects. The simulation of the memory experiments clearly evidences that the contribution of dipolar interactions is much stronger than that of exchange interactions.

Methods

Experimental techniques and data treatment

MnFe2O4 nano-powders have been synthesized by coprecipitation of Fe3+ and Mn2+ from water-in-toluene reverse micelle system and subsequent thermal treatment at 320°C. Detailed synthesis procedure, structural and morphological characterization are reported elsewhere18,19.

DC magnetization measurements were performed in zero-field-cooled (ZFC), field-cooled (FC), thermo-remanent (TRM) and isothermal remanent (IRM) conditions using a Quantum Design SQUID magnetometer equipped with a superconducting coil (Hmax = 5 T). The temperature-dependent ZFC, FC, TRM, and IRM magnetization measurements, as well as the magnetic field-dependent direct current demagnetization (DCD) and IRM are described in more details in the Supplemental Materials. To avoid any movement of the nanoparticles during the measurements, the samples, in the form of powders, were immobilized in epoxy resin. The ac-susceptibility χ data was recorded as a function of the temperature T and frequency ω = 2πf on the same system as well as on a Quantum Design physical property measurement system (PPMS).

Monte Carlo model

We have used the Monte Carlo (MC) simulations technique to calculate the memory behavior of the dense assembly of ultra-small Mn ferrite nanoparticles. A detailed description of our mesoscopic model that includes the core/surface morphology of each particle and the interparticle interaction is given in Ref.13. Here, in the Supplementary materials we have also included a brief description of this model in order to provide the parameters entering our simulations. In short, three spins (one core and two surface ones) are considered for each particle, yielding exchange interaction (with strengths Jc and Jsrf) and anisotropy terms (Kc and Ksrf). The interparticle interaction is considered by including exchange (Jinter) and dipolar (g) interaction. For the calculation of the Reference and Memory ZFC and TRM curves we follow the experimental procedure: (1) the system was first cooled at a constant temperature rate from T = 0.8 in zero field in the ZFC magnetization procedure and by applying a low field H = 0.08 for the calculation of the TRM curves. (2) hold-and-wait at temperature Th = 0.18 for a waiting time (ZFC and TRM) tw = 5 × 106 Monte Carlo Steps per Spin (MCSS). (3) Continue cooling down the system to the lowest temperature T = 0.002 (4) heating the sample in the presence of a magnetic field H = 0.08 for ZFC; for the TRM, the field was removed. The magnetization was monitored in the step (4) for each temperature. Detailed description of the model is reported in Supplementary Materials.