Dual disorder-driven magnetic dynamics in GdCu2 superantiferromagnetic nanoparticles

The spin dynamics in magnetically disordered GdCu2 nanoparticles, varying the nanoparticle size in the range 53 to 7 nm, has been scrutinized. Dynamic χAC susceptibility measurements have revealed the existence of dissipation at Tg = 18 K, which is associated to the spin freezing transition, for all the ensembles. Besides, the superantiferromagnetic ensembles (〈D〉≥ 24 nm) also showcase a dissipation contribution close to the vicinity of the Néel transition, TN = 40.2 K. This dissipation, which takes the form of two humps located at Td1 = 33.5(5) K and Td2 = 40.0(5) K, is associated to uncompensated antiferromagnetic moments. Time-dependent phenomena (ageing and memory effects) are only evidenced below the spin freezing transition, evidencing that solely this low-temperature disordered phase is driven by the frustration of RKKY exchange interactions. Consequently, GdCu2 nanoparticles display a dual disorder-driven magnetic dynamics, which are the one ascribed to the magnetically frustrated moments located at the nanoparticle surface; and that of uncompensated antiferromagnetic moments located within the nanoparticle core.


Introduction
Over the last years, the spin dynamics of magnetic ensembles, including mesoscopic ones, has received increasing interest. Their technological transfer, specially focused on the spintronics fields, has been the driving force motivating the experimental and theoretical efforts to clearly understand how the size reduction to the nanoscale affects the magnetic properties (Golosovsky et al. 2021;Baczewski et al. 1989;Jefremovas et al. 2020aJefremovas et al. , 2021Zhao et al. 2019).
Abstract The spin dynamics in magnetically disordered GdCu 2 nanoparticles, varying the nanoparticle size in the range 53 to 7 nm, has been scrutinized. Dynamic χ AC susceptibility measurements have revealed the existence of dissipation at T g = 18 K, which is associated to the spin freezing transition, for all the ensembles. Besides, the superantiferromagnetic ensembles (〈D〉≥ 24 nm) also showcase a dissipation contribution close to the vicinity of the Néel 1 3 Vol:. (1234567890) Within this context, antiferromagnetic (AF) nanoensembles have attracted a lot of interest in recent years, owing to their inherent assets (e.g., anomalous Hall effect, anisotropic magnetoresistance, longdistance spin-wave propagation (Marti et al. 2014;Nakatsuji et al. 2015;Lebrun et al. 2018). It is also worth mentioning that the complex magnetic arrangements found in AF materials makes them potential candidates for hosting exotic spin textures, as it is the case of skyrmions (Zhang et al. 2016;Fukami et al. 2020;Jungwirth et al. 2016;Jungfleisch et al. 2018;Everschor-Sitte et al. 2018;Park and Kim 2021). It is within this context that the determination of the spin dynamics governed by disordered and frustrated magnetic interactions is key, since the magnetic frustration arising from RKKY interactions have shown to be helpful in stabilizing these topologically protected spin textures (Greedan 2006;Zvyagin 2013;Tokura and Kanazawa 2020;von Malottki et al. 2017;Yuan et al. 2017).
Among the Rare-Earth binary alloys, the GdCu 2 bulk alloy displays an AF structure (Rotter et al. 2000b), which is maintained down to 24-nm-sized magnetic nanoparticles (MNPs), all along with the onset of a frustrated Spin Glass (SG) phase at low temperature (Jefremovas et al. 2020b). Among the R elements, Gd 3+ -ions display a half-filled 4f shell, implying absence of spin-orbit interaction. This leads Gd 3+ to be in an intermediate situation between that of 3d and 4f magnetism. The S-state of Gd 3+ also implies the lack of magnetocrystalline anisotropy for this ion, which certainly eases theoretical modeling and calculations. Thus, Gd 3+ has been more and more included in compounds to understand their magnetism at the basis, as it has been the case of the recent study reported by P.G. Welch et al., where the spin dynamics of the Heisenberg pyrochlore antiferromagnet Gd 2 Pt 2 O was unraveled (Welch et al. 2022). Furthermore, Gd 3+ displays, along with Tb 3+ , one of the greatest effective magnetic moments (μ eff = 7.94μ B ), which constitutes another asset to determine subtle modifications in the interaction among the magnetic moments. This has been useful, for instance, to reveal the existence of two-length scales reported for nanocrystalline Gd (Döbrich et al. 2009), or, more recently, to reveal how the RKKY magnetic interactions were altered by the size reduction (Jefremovas et al. 2020b). In this latter work, Jefremovas et al. have determined 24 nm to be the limit size for the bulk AF state to survive within the nanoparticle core, together with a surface SG phase, building the so-called superantiferromagnetic (SAF) state. The RKKY interactions of smaller nanoparticles are unable to establish a collective AF ordered state, and a Super Spin Glass (SSG) arrangement is settled instead. This lead GdCu 2 to display a particular magnetic order/disorder configuration depending upon the nanoparticle size. Indeed, a detailed study of the underlying spin dynamics is key to understand the modifications of the RKKY exchange interactions at the nanoscale.
Hence, in the present work, we carried out dynamic χ AC susceptibility measurements, both in the temperature and frequency domains (T,f), and in the time domain (t), to unravel the nature behind the magnetic order/disorder phases found in GdCu 2 MNPs. Complementary, specific heat c P measurements were analyzed to better determine the nature of the magnetic transitions. The present characterization is not so commonly found in the literature, and supplements the static picture of the magnetic properties of GdCu 2 MNPs reported in Jefremovas et al. (2020b).
The magnetic characterisation was performed by means of dynamic χ AC (t,f,T) (time, frequency and temperature, respectively) measurements. These were carried out in two QD-MPMS (SQUID) magnetometers, one of which is located at the Uppsala University (Sweden), and the second one, at the Universidad de Cantabria (Spain). The ensembles of MNPs were measured in different temperature ranges between T = 5-300 K. Oscillating fields μ 0 h = 0.1 and 0.313 mT, and frequencies ranging from 0.17 Hz to 2 Hz were employed. To probe memory effects and ageing phenomena, several protocols can be found in the literature (Nordblad and Svedlindh 1998;Jönsson et al. 2000;Jonason et al. 1998;Joshi et al. 2020;Svedlindh et al. 1992). Here, we have probed both time-dependent phenomena by tracing the out-of-phase ′′ component of the dynamic χ AC susceptibility, as it allows to detect in more detail the subtleties concerning the spin dynamics (Nordblad and Svedlindh 1998;Svedlindh et al. 1992;Jefremovas et al. 2022). Shortly, memory effects have been probed from the difference between the out-of-phase ′′ component measured (i) during cooling ( ′′ cooling ), and (ii) upon warming ( ′′ warming ). There is a difference between the measuring protocol used while cooling and warming. This way, during the cooling, a stop is made at T ag = 15 and 30 K for t > 10 3 s. During this time, the system gets aged, and a particular spin disorder configuration (domain) can be settled. Then, the warming curve is measured without making any stop. The memory effect is then evidenced by the occurrence of a drop in �� warming − �� cooling at temperatures slightly below T g . Coming to the ageing phenomena, these can be easily detected by the inspection of the ′′ vs. t dependency. Moreover, the robustness of the SG-like frustrated interactions has been further investigated by applying a temperature cycling protocol. This consists of measuring the ′′ vs. t dependency at a certain temperature T ag within the SG phase (in our case, T ag ∼ 0.8T g ) for a sufficiently long period of time ( t ∼ 10 3 s). After the waiting time, the temperature is raised to T ag + ΔT. In the case of the present work, ΔT was selected such that T ag + ΔT was 0.85T g and 1.1T g . Then, the temperature is lowered back to T ag , and �� (t) is measured again for t ∼ 10 4 -10 5 s. This cycling protocols mimics the one reported in ?svedlindh1992time,ageing_Eli ().
Finally, heat capacity (c p ) measurements were performed using a QD-PPMS instrument (Universidad de Cantabria) in the temperature range T = 2-300 K. The measurements were collected in the absence of magnetic field. Measurements were performed following the relaxation method (Bachmann et al. 1972), and the data analyses follows the surface-core separation already detailed in Jefremovas et al. (2021Jefremovas et al. ( , 2022.

Results and discussion
Dynamic magnetic susceptibility vs. temperature Figure 1a and b showcase the χ AC (T) components (inphase ′ and out-of-phase ′′ , respectively) measured at the frequency of f = 0.17 Hz for the 6 ensembles of MNPs. There, the occurrence of a cusp in both in-phase ( Fig. 1a) and out-of-phase components ( Fig. 1b) is clearly noticeable in the low-temperature region (T < 25 K). This has been interpreted as the onset of a SG-like phase, whose freezing dynamics has been characterized in great detail in Jefremovas et al. (2020b). Therefore, from herein, we will focus on the high temperature region, i.e., the one close to the Néel transition.
Therefore, in Fig. 1a, a peak associated with the AF transition is found at T N = 40.2 K for MNPs larger than 24 nm, being absent for the smallest ones (13 and 9 nm). The temperature value of this peak does not evidence a size-dependence, as expected (Coey 2010). Paying attention now to the out-of-phase component displayed in Fig.1b, additionally to the onset of a low-temperature peak (T g ≈ 18 K), linked to the SG phase, a dissipation contribution is clearly detectable in the temperature range between 30 and 40 K, but only for the case of 〈D〉≥ 24 nm MNPs. More precisely, this dissipation takes the form of two humps, located at T d1 = 33.5(5) K and T d2 = 40.0(5) K for all these superantiferrromagnetic (SAF) MNPs. The peaks broaden and reduce in magnitude with the size reduction, being completely wiped out when the limit size for AF order to remain is overcome (i.e., below 24 nm). The observation of dissipation connected to AF order is totally unexpected, as these should not exhibit, in principle, any contribution to the out-of-phase component (Coey 2010). There is, however, one scenario where dissipation can be expected for AF MNPs, and it is found within the context of uncompensated magnetic moments associated to antiferromagnetic domains. This interpretation follows the same ideas previously proposed for NiO AF grain boundaries (Takano et al. 1997), and it is congruent with the static M DC (H) characterization described in Jefremovas et al. (2020b), where the evolution of the coercive field H C with MNP size 〈D〉 reached a maximum for 24-nm-sized MNPs. A simple estimation based on the AF unit cell, which comprises 3 crystallographic-ones along b direction (Rotter et al. 2000a), leads to an AF correlation length of 3 b ∼ 2.1 nm. Hence, the 24-nm-sized MNPs, for which a core size of ⟨D⟩ ∼ 20 nm can be estimated (Jefremovas et al. 2021(Jefremovas et al. , 2022Rojas et al. 2007), are large enough to host up to 11 complete AF unit cells and the AF grain boundaries within. The fact that this dissipation takes the form of two well-defined and separated cusps is in further agreement with the AF structure of GdCu 2 .
According to neutron diffraction and muon spectroscopy resonance analyses (Rotter et al. 2000a;Gygax et al. 2002;Rotter et al. 2000b), GdCu 2 arranges into a commensurate AF structure carrying non-collinear cycloidal propagation. There are two different domains possible, one with a left-handed and another with a right-handed cycloid. Each of these two domains should carry frequency-dependent dissipation, a fact that can be observed in Fig. 1c for the 33-nm-sized MNPs, as there is a progressive decrease of the ′′ component at T d1 and T d2 with increasing f. Furthermore, the trace of a high-temperature double-peak signature is also traced by the nonlinear ′ 3 response, included in Fig. 1d. The lack of these two peaks also in the non-linear response corresponding to the SSG MNPs (13-and 9-nm-sized) necessarily connects this high-temperature dissipation to the AF structure.

Time-dependent phenomena: memory effects and ageing
Ageing and memory effect phenomena are intimately connected to the out-of-equilibrium dynamics of nonergodic systems (SGs), thus proving the existence of highly correlated RKKY-frustrated spins (Svedlindh et al. 1992;Nordblad and Svedlindh 1998 Binder and Young 1984;Jonsson et al. 1995;Mydosh 2014). Figure 2a depicts the temperature dependence of out-of-phase component, measured while cooling down and letting the system stay at T ag = 15 and 30 K for t > 10 3 s; and upon warming without making any stop. The inspection of the �� warming − �� cooling difference showcased at the bottom inset allows to clearly detect memory effects below T < 15 K, as a drop in the difference is observable at temperatures below T ag = 0.83 T g = 18 K. This effect, which is triggered by the out-of-equilibrium dynamics ascribed to the freezing transition, shows up for all MNPs except for the 53-and 43-nm-sized ones. The lack of memory effects at the larger MNPs reveals that the AF-coupled core is still robust towards the magnetically frustrated surface phase. By comparing the drop in the �� warming − �� cooling difference, two dif-ferent trends can be deduced: (i) for the SAF MNPs (33 and 24 nm), the size reduction seems to reduce the memory effects, as the �� warming − �� cooling difference is broader for the 18 nm ensemble as compared to the 33-nm-sized one. This is congruent with the stated more robust SG phase of the 33-nm-sized MNPs (Jefremovas et al. 2020b), as the RKKY exchange interactions are less affected by the microstrain. Then, once the AF order is destroyed, and a SSG state is settled, (ii) memory effects get more intense as the nanoparticle size is reduced. Accordingly, the SSG MNPs (9-nm-sized) display stronger memory effects than those showcased by the 13-nmsized ones. Both SSG ensembles display stronger memory effects compared to the SAF ensembles. This can be deduced from the sharpness of the This drop is displayed in the bottom inset, which also includes the difference corresponding to 24-, 13-and 9-nm-sized MNPs. The �� warming − �� cooling is divided by the respective ′′ Tw for each nanoparticle size to compare the drop accounting for the memory effects. The top inset displays the ′′ vs. t curves recorded at T ag = 30 K. b) and c) display the cycling protocol results of 33-nm-sized and 9-nm-sized MNPs, respectively. The relaxation is measured with f = 0.2 Hz at T ag = 15 K before and after the temperature cycling. It can be seen that the SG state is completely reborn when the rise is of T ag + ΔT = 1.06T g in both MNP ensembles. Inset in c) compares the relaxation measured at T ag = 12, 15 and 18 K for the 9-nm-sized MNPs the top-right inset in Fig. 2a evidences the lack of a clear time-dependence associated to the high temperature dissipation (compare with Fig. 2b and c, where the time-dependence obeys the expected decay for spin glasses at several T ag < T g ). This backs up the idea of a non-frustrated disordered phase, which is congruent with uncompensated AF moments, as it has been explained in the previous section.
In order to test the robustness of the SG phase, temperature cycling have been performed. To this aim, the MNPs were aged for t ∼ 10 3 s at T ag = 15 K. Then, the temperature was rose to T ag + ΔT ≈ 0.88, 0.97 and 1.06 T g and cycled back to T ag . Immediately, the magnetization was recorded for t ∼ 10 4 s. Figure 2b and c show results for these measurements performed in 33 nm (SAF) and 9 nm (SSG) sized MNPs, respectively. As it can be seen, both MNP ensembles achieve a completely reborn SG landscape (rejuvenation) when the cycling step is performed above T g (T + ΔT ≥ 1.06T g ), as the ′′ post-cycle (t > 10 3 s) is equal to the former ′′ pre-cycle (t = 0s). Also, in both MNP ensembles, the smaller the ΔT, the slower the relaxation towards equilibrium, which indicates that larger free-energy barriers are built. This reveals that the domains of correlated spins are larger (Jonason et al. 1998).
If we stick close to the rejuvenation limit by paying attention to the cycle performed at T + ΔT = 0.97T g , it can be observed how, despite the different global states (SAF and SSG), the SG freezing dynamics corresponding to 33-and 9-nm-sized MNPs behave in a very similar fashion. In this way, the �� ∕ �� (t = 0) value at both 33 nm and 9 nm MNPs after the temperature cycle is around �� ∕ �� (t = 0) = 0.98, whereas it is already �� ∕ �� (t = 0) = 1 (fully recovered) for 24and 13-nm-sized ones (not shown). This implies that the domains built for the former (33 and 9 nm) are larger, as the particular SG configuration is not completely reborn after the cycling. The reason beneath this feature further supports the static picture, from which it was deduced that the most interacting SG phase is settled for 33-nm-sized MNPs (Jefremovas et al. 2020b). Once the SSG state is established, the smaller the MNPs, the more robust the frustrated interactions among the spins.

Specific heat measurements
Specific heat analyses have been carried out separating the contributions stemming from the core and the surface, following the same procedure as the one published in Jefremovas et al. (2021Jefremovas et al. ( , 2022. To this aim, the experimental specific heat c P is assumed to be the result of the lattice contribution c lattice (formed by the addition of the electronic and the phononic ones, i.e., c el + c ph ), plus the magnetic c mag . In the case of GdCu 2 , since Gd 3+ are S-state ions, no crystalline electric field (CEF) contribution is present in the GdCu 2 specific heat. Each core and surface contributions are weighted by the geometrical core and surface-to-volume ratios, respectively, which have been estimated in the same way as in precedent works (Jefremovas et al. 2021(Jefremovas et al. , 2022. Accordingly, the core and surface contributions are N c = 2.0 (33 nm), 1.0 (13 nm) and 0.9 (9 nm), and N s = 3 − N c , respectively, as the number of atoms is N = 3). Therefore, the specific heat is modeled following: Following this procedure, the green line in Fig. 3a represents the obtained c lattice for 33-nm-sized MNPs. Values of γ c = 6.7 (2) mJ (molK 2 ) -1 and c D = 277(3) K have been obtained for this particular size, which agree well with the ones obtained for bulk alloy (not shown), and reported for polycrystalline bulk GdCu 2 (Podgornykh and Kourov 2007). The obtained γ s and s D values for all the MNP sizes are listed in Table 1. There, it can be seen that both parameters increase with the size reduction, a fact that can be understood according to the increasing surface disorder and surface-to-core ratio.
The inset of this figure shows the magnetic entropy S mag against the temperature. This S mag has been obtained according to: The experimental magnetic entropy (18 J/mol⋅K) and the calculated S theo mag (300K) = R[ln(2J + 1)] = 17.29 J/mol⋅K are in good agreement. The value of S exp mag is already ≈ 18 J/mol⋅K at T = 100 K. This indicates that the energy levels are fully populated at a lower temperature than expected. This trend, which holds for all the GdCu 2 MNPs, may be indicative of the existence of quadrupolar and/or higher order interactions (Luong and Franse 1995;Morin and Schmitt 1990).
(1) Figure 3b depicts the c mag vs. T dependency for bulk, 33-, 13-and 9-nm-sized MNPs. It is worth noting the two sources of c mag evidenced by the bulk and 33-nm-sized alloys. Accordingly, the AF-coupled magnetic moments give rise to a λ-like peak anomaly, located at T ∼ 40 K. This contribution should show a slight left-shift with the size reduction, owing to the reduction of the amount of AF-coupled moments (García-Saiz et al. 2014;Chevalier et al. 2006). Nevertheless, no shift is found in our GdCu 2 alloys, as it was also the case of the M DC (T) curves shown by Jefremovas et al. in (2020b). This underlines, in agreement with the dynamic χ AC (t,f,T) characterization, the robustness of the AF order. Additionally to the AF λ-anomaly, at temperatures below T N , a broad hump, which can be ascribed to a Schottky-like contribution (Gopal 2012), emerges for these bulk and 33-nm-sized MNPs. In the present case of GdCu 2 , the occurrence of this Schottky contribution should be ascribed to the Zeeman splitting of the eightfold degenerate energy level, rather than to CEF effects (as Gd 3+ is L = 0) or spin wave propagation (Luong and Franse 1995;Luong et al. 1985). Furthermore, the fact that the excess of c mag drops to zero for T > T N rules out the possibility of a CEF-motivated contribution to c P (Gopal 2012), and it is congruent with the results reported for GdCu 2 single-crystal (Koyanagi et al. 1998) and for GdCu x bulk antiferromagnets (Podgornykh and Kourov 2007). Both contributions (AF λ-anomaly and Schottky contributions) are so close to each other that they overlap, resulting in a single broad cusp, rather than in two separated signatures.
On the other hand, the c mag (T) dependency of the SSG 13-and 9-nm-sized MNPs evidences a broad cusp at T ∼ 31 (13 nm) and T ∼ 27 K (9 nm), with a tail (asymptotic decrease to zero of the c mag ) that extends up to T ∼ 100 K. The cusp intensity is reduced with respect to the one of bulk and 33 nm MNPs, and it is also found at lower temperatures. The interpretation of the c mag of these SSG MNPs is very alike to the one already commented for bulk and 33-nm-sized MNPs. Even though the AF interactions are not strong enough to give rise to a collective well-defined ordered state within these SSG MNPs, they still exist within the sample, as they are a basic requirement for magnetic frustration (Mydosh 2014   of the MNPs give rise to an effective local field, which splits the multiplets (Zeeman splitting), resulting in a contribution to c mag . Of course, as the AF order interactions are further diminished by reducing the MNP size (increasing disorder), this splitting gets smaller, thus, the hump shifts towards lower temperature value. Finally, the c mag asymptotically decreases to zero, a fact that contrasts with the drop observed for bulk and 33-nm-sized MNPs. This tail shall be ascribed to the SG-frustrated moments, which may give rise to (tiny) contributions to the specific heat at higher temperatures (Martin 1979;Mydosh 2014;Gopal 2012;Arons et al. 1994).

Conclusions
The dual magnetic disorder dynamics in ensembles of GdCu 2 magnetic nanoparticles (53-to 7-nm-sized) has been unraveled. On the one hand, at low temperature ( T g ≲ 18 K), an interacting SG phase is well-established, evidencing rejuvenation and memory effect phenomena. This frustrated phase comes as a result of the size reduction to the nanoscale, driven by (i) the surface frustrated moments for the case of SAF nanoparticles (53-, 43-, 33-and 24-nm-sized), and (ii) the whole nanoparticle, for the case of SSG-ones (13 and 9 nm). On the other hand, a high temperature ( T d1 ∼ 33 K and T d2 ∼ 40 K) non-frustrated disordered phase has been found only for the SAF nanoparticles. This dissipation, which does not evidence a time dependency of the AC susceptibility, and therefore magnetic ageing, is not present for the SSG nanoparticles. This necessarily connects the high-temperature dissipation to a disorder driven by the uncompensated AF moments, found at the AF grain boundaries. The observation of two cusps for this hightemperature disorder is intimately related to the cycloidal propagation of the GdCu 2 magnetic structure, which includes left and right-handed domains. Our results demonstrate the importance of ageing and specific heat as powerful tools to unravel the subtleties concerning the spin dynamics in magnetic GdCu 2 nanoparticles.
Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work has been financially supported by Spanish MCIU MAT2017-83631-C3-R project. EMJ work was supported by 'Beca Concepción Arenal' BDNS: 406333 (Gobierno de Cantabria-Universidad de Cantabria).

Data availability
The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.

Conflict of interest
The authors declare no competing interests.
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