Critical magnetic behavior in [Ag 8 /Co 0.5 ]x64, [Ag 8 /Co 1 ]x32 and [Ag 16 /Co 1 ]x32 epitaxial multilayers

We investigate the low temperature magnetic behavior of three epitaxial Co/Ag multilayers, grown onto MgO (001) substrates, with a nominal content per period of either half a monolayer or one monolayer of Co, and either 8 or 16 Ag monolayers. The samples were studied by X-ray reﬂectivity and diffraction, transmission electron microscopy, magnetometry and ac susceptometry. The results indicated a well deﬁned stacking sequence in the growth direction, the number of periods and of Ag monolayers per period being coincident with the nominal values for each sample. The Co layers were found to be discontinuous and corresponded to a quasi-monodisperse in-plane distribution of Co nanoparticles embedded in a Ag(001) matrix. The zero-ﬁeld cooled and ﬁeld cooled temperature variations of the low ﬁeld magnetization indicated the presence of irreversibilities at temperatures below 20 K. The ac ﬁeld frequency (f) and temperature (T) dependencies of the real part of the susceptibility ( χ ′ ) corresponded to a Vogel–Fulcher behavior in the three samples, and indicated a frequency shift parameter ( Γ ) of the order of 4 x 10 -2 . For each sample, the experimental data corresponding to the variations of the imaginary part of the ac susceptibility ( χ ′′ ) with f and T were found to collapse into a single curve according to the dynamic scaling law. Taken together, these results allow us to conclude that the three multilayers experience a phase transition of the paramagnetic to superspin glass type, driven by the dipolar interactions between the Co nanoparticles. Regarding the inﬂuence of the multilayer features, we found a clear dependence of the order parameter of the transition on the nominal number of Co monolayers per period.


INTRODUCTION
Quenched disorder and competing interactions coexisting in a system constitute the necessary ingredients for the occurrence of the so-called spin glass behavior. 1 That behavior is largely ubiquitous and has been identified in many different fields, 1 like the analysis of the human brain functionality, 2 neural networks, 3,4 prebiotic evolution 5 or protein folding. 6 Nevertheless, the paradigm of the spin glass phenomenology and of its theoretical understanding are magnetic moments systems including distributed interactions. Actual realizations of magnetic spin glass systems include from the simple canonical spin glass metallic phases (where diluted, randomly distributed, localized moments interact through the spatially oscillatory RKKY interaction), to more complex magnetic materials incorporating different inter-moments interactions (from superand double-exchange to dipolar coupling). The magnetic spin glass systems also encompass a broad range of dimensionalities, disorder types and disorder parameters distributions. 1,7,8 In a previous work 9 ARTICLE scitation.org/journal/adv on a Co/Ag multilayer with granular morphology at the Co layers, we have shown the occurrence at low temperature of a superspin glass phase transition governed by the dipolar interactions between the freezing Co nanoparticles. Here we extend the analysis of the low temperature magnetic behavior to a series of Co/Ag multilayers also including a reduced Co content per period and explore the influence of varying the Co or Ag content per period.

SAMPLE PREPARATION AND EXPERIMENTAL TECHNIQUES
The three multilayers ( Samples were covered by a 3 nm Ag capping layer, and, in the following, will be termed X/Y, X being the nominal number of Co monolayers (ML) per period, and Y that of Ag ones. Here X is either 0.5 or 1 Co ML, while Y is either 8 or 16 Ag ML. In order to deposit the same total nominal number of Co atoms per surface unit at the three samples, the period was repeated 32 or 64 times depending on the X value (1 or 0.5). The stacking, structure and morphology of the samples were investigated by means of X-rays reflectivity (XRR), diffraction (XRD) and scanning transmission electron microscopy (STEM). The magnetic characterization was performed by using vibrating sample and SQUID magnetometers (fields of up to 90 kG) and an ac susceptometer, covering the range of ac field frequencies (f) from 10 −1 up to 10 4 Hz, and temperatures (T) from 2 K up to 290 K.

TABLE I.
Values of different parameters derived from the magnetic analysis for each sample (details in the text). T irr : irreversibility temperature observed at the ZFC/FC low field magnetization curves; Vpara and Øpara: paramagnetic average volume and diameter estimated for the Co nanoparticles from the Curie-Weiss fits; Γ: frequency shift parameter; T 0 and Ea/k B : magnitude of the interactions and activation energy for system relaxation, respectively (Vogel-Fulcher model); Zν (critical exponent) and Tg (phase transition temperature for an infinite observation time) obtained from the fit of the ac susceptibility to the scaling law at the critical regime.

RESULTS AND DISCUSSION
The XRD, XRR, and STEM results indicated that the MBE deposited films exhibited good crystallinity, fcc(001) structure and clear superlattice periodicities, the number of periods in each sample being coincident with the nominal ones (see e.g., the XRR data shown in Figure 1a for the 1/8 and 1/16 samples). The STEM images also revealed that, as reported inRef. 9, the Co layers were not continuous (see Fig. 1b), but exhibited a close-to-monodisperse inplane distribution of Co nanoparticles (NPs) embedded in a Ag(001) matrix. The characteristic size and shape of the Co NPs for the 1/16 sample can be seen in Figure 1b, which also shows the presence of 16 Ag MLs between two consecutive Co layers. The STEM analysis of the different samples indicated a disordered distribution of the Co NPs positions within each layer, and confirmed the nominal Ag periodicity. For the 1/8 and 1/16 samples, the distance between the lateral surfaces of two neighboring NPs in a Co layer is of the order of 1.5 nm, similar to the average NPs diameter.
The zero field-cooling (ZFC) and field-cooling (FC) variations with T of the dc low field magnetization indicated the occurrence of irreversibilities at temperatures (T irr ) lower than 20 K (see Fig. 1c for the 1/8 sample, and T irr values for each sample in Table I). Below T irr , the multilayers exhibited hysteresis with dc coercivities of up to 350 Oe at 2 K. Above T irr , the ZFC initial dc susceptibility (χ) behaved according to the Curie-Weiss law, 10 1/χ = (T -θ)/C, in the three samples. The values obtained from the fits to the Curie-Weiss law yielded paramagnetic moments two orders of magnitude larger than the Co atomic moment. They approximately correspond to the clustering of fcc coordinated Co moments in a NP with a diameter between 1 and 2 nm. Since this is consistent with the granular morphology of the Co layers revealed by STEM, these Curie-Weiss paramagnetic moments can be associated to the Co NPs and we used  Table I. Figure 2 shows the temperature dependencies of the real (χ ′ ) and imaginary (χ ′′ ) parts of the ac susceptibility 1 measured at ac field frequencies (f) in the range from 1 Hz up to 10 kHz. Maxima on χ ′ are observed at temperatures (T p_ac ) of the order of T irr . T p_ac and their associated peak magnitudes increase and decrease, respectively, as f increases. The variations of T p_ac , with f are parameterized 1 by considering the relative increase of T p_ac per decade, according to Γ = (Δ T p_ac /T p_ac )/Δ log 10 f, and taking as reference f = 1 Hz. The Γ values thus obtained (see Table I) are clearly lower than those associated to superparamagnetic relaxation processes (which in a large majority of materials are above 0.2 1 ). These results suggest that, like in our previous report, 9 the magnetic properties of these multilayers are associated to the Co NPs of the nominal Co layers, and that such NPs could exhibit at T irr a phase transition from a paramagnetic to a magnetically ordered sate.
In order to assess the actual occurrence of interactions between the Co NPs, we have fitted the experimental data for the frequency variation of Tp_ac to the Vogel-Fulcher model. 11 That model describes the temperature dependence of the characteristic times of the spin glass-like freezing 12 according to ω = ω 0 exp [-E a /k B (T p_ac -T 0 )], where ω is the ac field angular frequency and ω 0 , E a , and T 0 fitting parameters: ω 0 is the try frequency, E a the activation energy for the relaxation, and T 0 the magnitude (in temperature units) of the interactions present in the system; Table I shows the T 0 and E a /k B values obtained from the fits. For the three samples, ω 0 values are in the range of 5 x 10 9 s -1 , whereas T 0 values are of the order of the corresponding T irr . Thus, these results clearly endorse the presence of interactions whose magnitude could be on the origin of an ordered magnetic state at temperatures below T irr , transitioning to a superparamagnetic phase above T irr .
To analyze that behavior we will consider two different approaches. First, we will study the relationship of the temperature relaxation time τ(T) with the T p_ac experimentally determined values. 1,13,14 That relationship can only be stablished at the critical state, that is, at T near the phase transition at which the distribution of the magnetically correlated regions (through the interactions present in the system) spans all the possible dimensions compatible with the elemental interacting entities and the system sizes. The lifetimes of the correlated regions should also reach the characteristic time of the measurement. In this critical state, the spin relaxation time diverges with the reduced correlation length (ξ) according to τ = τ * 0 ξ Z (where Z is the dynamic scaling exponent). ξ is related to T p_ac through the law ξ = ε -ν (where ν measures the divergence of the correlation length, and ε = [(T p_ac /T g )-1]; T g being the temperature at which the phase transition occurs when the measuring time is infinite. In terms of the frequency f, the scaling law takes the form: f = f * [(T p_ac /T g )-1] Zν or equivalently, ln f = ln f * -Zν ln [(T p_ac /T g )-1] (Zν is the exponent linked to the collective state occurring below T g ). Figure 3a) displays, for the three multilayers, the dependence of the logarithm of the frequency f on the

ARTICLE
scitation.org/journal/adv quantity ln[(T p_ac /T g )-1]; the f * values considered for the three samples were of the order of 10 9 s -1 ; those of the Zν and T g parameters are shown in Table I. As it is readily seen, in all the cases there is a good agreement between the experimental data and the scaling law.
It is interesting to remark that the T g values are slightly lower than the T irr ones (as expected from the finite measuring times). Also, and importantly, the values of the critical exponent here obtained (varying from Zν = 6.7 to 9.6; see Table I) are clearly within the range identified in the literature as corresponding to the occurrence of spin glass-like systems freezing. [15][16][17] Our second approach to elucidate if a freezing transition occurs in these samples at T irr, is based on the dynamic scaling of the imaginary (absorption) component of the ac susceptibility (χ ′′ ). 18 That scaling predicts the collapse of the experimental data corresponding to the f and T variations of χ ′′ , in a single curve G(x), verifying: T χ ′′ (f, T) = [(T/T g ) -1] β G(x); in this expression x = f[(T/T g ) − 1] −Zν (where Zν is the critical exponent and T g is the transition temperature at zero frequency); β is the order parameter of the transition. 1 Figure 3b) displays the scaling behavior obtained for our three samples, which is in agreement with the existence of a freezing transition at temperatures close to T irr . Note that the β values, obtained from the collapse in Figure 3, are in the range corresponding to spin glass-like systems. 18 Interestingly, in these samples the β values are only found to vary with the nominal number of Co monolayers per period.

CONCLUSIONS
The above results allow us to conclude that the three multilayers experience a phase transition of the paramagnetic to superspin glass type at temperatures T irr . The freezing entities are the Co nanoparticles present at the nominal Co layers. The magnitude of the moments of these Co NPs, as well as that of the average interparticle distance, indicate the occurrence of high dipolar interactions. We thus propose that such interactions, together with the in-plane disorder of the NP positions, are on the origin of the spin glass-like magnetic order. 9 The size and in-plane concentration of the Co NPs can be also correlated with the variations observed in T irr depending on the nominal number of Co and Ag monolayers per period. It can be seen that the 0.5/8 sample exhibits a lower T irr than the 1/8 and 1/16 samples; this is due to the smaller volume and in-plane concentration of the Co NPs present in the 0.5/8 sample, and the consequently lower magnitude of the interparticle dipolar interactions. On the other hand, our results indicate that the 1/8 sample experiences the phase transition at a higher temperature than the 1/16 sample; this can be explained by considering the thickness of the Ag spacer, which for the 1/16 sample is large enough to significantly reduce the interlayer Co NPs interactions; in contrast, those interlayer interactions measurably contribute to the phase transition in the 1/8 sample. Finally, the values of the β exponent in the scaling relationship are only found to vary (in these samples) with the number of Co monolayers per period.