Disorder induced critical phenomena in magnetically glassy Cu-Al-Mn alloys

Measurements of magnetic hysteresis loops in Cu-Al-Mn alloys of different Mn content at low temperatures are presented. The loops are smooth and continuous above a certain temperature, but exhibit a magnetization discontinuity below that temperature. Scaling analysis suggest that this system displays a disorder induced phase transition line. Measurements allow to determine the critical exponents $\beta=0.03\pm 0.01$ and $\beta \delta = 0.4 \pm 0.1$ in agreement with those reported recently [Berger et al., Phys. Rev. Lett. {\bf 85}, 4176 (2000)]

Beyond equilibrium, the response of a physical system to an external applied force shows hysteresis. Close to the rate independent limit, when thermal fluctuations do not control the kinetics of the system, the hysteresis path is reproducible from cycle to cycle. This athermal behavior is observed, for instance, in the magnetization reversal through a first-order transition in many magnetic systems at low enough temperatures. Modeling the shape of the hysteresis loops is a difficult problem that has attracted a great deal of interest for many years. Theoretical studies of the zero-temperature (absence of thermal fluctuations) hysteresis in a number of lattice models with different kinds of quenched disorder (random fields [1], random bonds [2], random anisotropies [3], etc...) have shown that by increasing the strength of disorder, the hysteresis loops change from displaying a discontinuous jump in the magnetization reversal (ferromagnetic character) to behave smoothly with the applied field. This change has been interpreted as a disorder induced non-equilibrium phase transition. For this phase transition, critical exponents corresponding to different models have been computed from numerical simulations and renormalization group methods. Moreover, universal scaling has been predicted. For instance, for short-range 3d models [4] β = 0.035 ± 0.028 and βδ = 1.81 ± 0.32, whereas for mean field models [5] β = 1/2 and βδ = 3/2.
In contrast to the considerable amount of theoretical research, only very few experimental works have been devoted to corroborate the actual existence of disorder induced phase transitions in the hysteresis loop. Obradó et al. [6] showed that the hysteresis loops in the magnetically glassy phase of Cu-Al-Mn change with increasing disorder (related to the alloy composition) as predicted by the models. However, no critical exponents were estimated in that work. Later, Berger et al. [7] studied the effect of magnetic disorder on the reversal process in Co/CoO bilayers. In this case, the antiferromagnetic CoO layer allows a reversible tuning of dis-order by temperature variation. They prove the existence of hysteresis-loop criticality and determine critical exponents. Nevertheless, the obtained exponents (β = 0.022 ± 0.006 and βδ = 0.30 ± 0.03) show important discrepancies with the theoretical predictions. Therefore, more effort is needed in the experimental analysis of other candidates in order to confirm the existence of disorder induced phase transitions as predicted by theory.
In this letter, we present magnetic hysteresis loops for different compositions and temperatures below the freezing temperature in a Cu-Al-Mn alloy. A critical transition line associated with the onset of a discontinuity in the hysteresis loop in the temperature-field-composition space has been determined, and the critical exponents have been obtained from the scaling of the magnetization curves. Our exponents show a remarkable agreement with the previous experimental estimates [7], thus supporting the idea of universality in such disorder induced transitions. Nevertheless, results also indicate that T = 0 theoretical models may lack an essential physical ingredient in order to account for finite temperature measurements.
A Cu 66.6 Al 20.4 Mn 13.0 single crystal has been grown by the Bridgeman method. The crystallographic structure is L2 1 (F m3m)which can be viewed as four interpenetrated fcc sublattices [6]. From the ingot, a parallelepiped specimen (a ≃ 3.4 mm, b ≃ 2.4 mm and c ≃ 2.2 mm) has been cut with a and b oriented along the [001] and [110] directions, respectively. The sample was annealed for 10 min at 1080 K, quenched in a mixture of ice and water and aged at room temperature for several days. The magnetic properties of Cu-Al-Mn arise from localized magnetic moments at Mn-atoms. These moments are coupled through an oscillating effective RKKY interaction. Due to non-stoichiometry, Mn-atoms are not uniformly distributed on one of the L2 1 sublattice sites. This disorder is at the origin of the glassy behavior observed at low temperatures [8]. The magnetization as a function of temperature and magnetic field has been measured with a SQUID magnetometer. Results presented here have been corrected for demagnetizing field effects. Fig. 1 shows the behavior of the zero field cooled (ZFC) and field cooled (FC) susceptibilities as a function of temperature. Measurements have been done by applying a field of 100 Oe. The different symbols correspond to the field applied in the [001] and [110] crystallographic directions. No significant anisotropy is observed. The ZFC and FC curves split at a temperature of 43 K below the position of the χ peak at ∼ 55 K. The splitting confirms the onset of irreversibility and the glassy behavior at low temperatures. Moreover, the inset shows an enlarged view of the FC susceptibility in this glassy region. The existence of a minimum in this curve indicates that, along this metastable path, there is a tendency for the system to develop a field induced ferromagnetic component at low temperatures. No indication of such a tendency can be observed in the ZFC curve. Fig. 2 shows a series of examples of hysteresis cycles at selected temperatures in the range from 5K up to 45K. Each loop has been obtained by cooling from 150 K in the absence of an external field down to the selected temperature and slowly varying the field from 0 → 50 kOe→ −50 kOe→ 50 kOe. This allows to obtain the virgin magnetization curve and the full saturation loop. Although it is not displayed in the figure, above 40 kOe (and below −40 kOe) the loops show already a reversible behavior. Nevertheless, saturation of the magnetization is still not reached at 50 kOe. To a high degree of accuracy, the loops are symmetric under inversion of the magnetization and field axis. Note first that above T ≃ 43 K no hysteresis effects are observed, consistently with the behavior of χ(T ) in Fig. 1. Secondly, the loops in the range from 15 K to 45 K are typical of a glassy system while, at T < 15 K the loops display a jump associated with the field induced ferromagnetic component. This becomes more pronounced as the temperature is lowered. On the other hand, the virgin curves do not show any signature of a ferromagnetic component, in agreement with the behaviour of the ZFC susceptibility in Fig. 1. The characterization of the onset of the ferromagnetic component is presented in Fig. 3, where we show the behavior of the estimated coercive field H coe (T ) and the magnetization jump ∆M (T ) as a function of temperature [13]. The coercive field exhibits a peak at T c ≃ 13 K which, within the errors, coincides with the temperature where ∆M vanishes and is close to the position of the minimum in the FC susceptibility. These results corroborate the existence of a continuous phase transition which occurs within the glassy phase. In order to understand the role played by the configurational disorder and/or thermal fluctuations in this transition it is important to study results corresponding to other compositions for which the different stoichiometry give rise to different degrees of disorder. Besides the measurements presented above corresponding to x = 13.0%, we have used data for polycrystalline samples with x = 6.3% and x = 9.0%, published previously [8].
To characterize the critical behavior we have performed a scaling analysis of the M (H) curves. The direct fitting of the β exponent from the ∆M vs. T data (in Fig. 3) does not provide reliable estimates of β since the values of ∆M are strongly influenced by the numerical procedure used in the extrapolations [13]. For a fixed Mn content, the distance to the critical point is measured through the reduced temperature t = (T − T c (x))/T c (x) and reduced field h = (H − H coe (x, T ))/H coe (x, T ). The standard scaling hypothesis [1,7] for the positive and negative hysteresis loop branches is M ± = t β M ± h/t βδ . In order to scale data for different compositions, we generalize the preceding expression to include the fact that the magnetic moment µ(x) depends on the Mn content. Renormalization of the magnetization by a factor µ(x) yields the following scaling hypothesis: This hypothesis is tested in Fig.4 The obtained values, except for a multiplicative factor, are µ(x = 6.3%) = 0.566, µ(x = 9.0%) = 0.575, and µ(x = 12.9%) = 0.625. The numerical estimations of T c (x), obtained from the improvement of the collapses in Fig. 4, are T c (x = 6.3%) = 0.5 K, T c (x = 9.0%) = 6.0 K and T c (x = 13.0%) = 13.2 K. It is important to note that the collapse of the data corresponding to x = 13.0%, considered separately from the other compositions (without the need of fitting µ), render values of β and βδ within the same error bars.  An independent method to check the consistency of the scaling is to analyze the critical behavior of the H coe (x, T ) surface. According to the Clausius-Clapeyron equation, assuming the standard scaling behavior for the entropy and the magnetization and using Griffiths equality [9], one obtains Integration The consistency of the data with this predicted power-law behavior is checked in the inset of Fig. 4. The straight lines have a slope of βδ = 0.4. For the x = 13% case, for which the amount of data is larger, we can perform a linear fit of such a behavior, rendering H c (x = 13.0%) = 560 ± 15 Oe. The fit is shown in Fig. 3. Fig. 5 shows the complete (H, T, x) metastable phase diagram. The line across the (x, T ) plane is an estimation of the freezing temperature obtained from the susceptibility maxima (from Fig.1 and from Ref. 8). To a good approximation, this temperature determines the spin freezing temperature in the low concentration region. However, for concentrations larger than 8%, as can be seen in Fig. 1, the position of the peak overestimates the temperature where irreversibility occurs. The thick line is the line of critical points, at the edge of the first-order transition surface (shaded) where the magnetization exhibits a discontinuity. The projection of this critical line is also shown on the H = 0 plane.
The exponents β and βδ obtained in the present work are in good agreement with those reported recently in the study of Co/CoO bilayers [7]. In that case, the authors argue that their system exhibits a disorder induced phase transition. The amount of disorder is controlled through the antiferromagnetic order of the CoO layer which changes when the temperature crosses the Néel point. In our case we can guess a similar origin, since it is know that ferromagnetism and antiferromagnetism coexist in non-stoichiometric Heusler alloys [6,10,11]. This hypothesis is also consistent with the behavior of the hysteresis loops that become closed at relatively small fields but do not saturate up to high fields. We suggest that even at the low temperatures studied in the present work, the antiferromagnetically interacting moments act as random fields on the frozen ferromagnetic moments. Temperature changes are able to alter this effective disorder causing the disorder-induced phase transition. We want to note that the zero temperature model for Cu-Al-Mn [8], which considers the random position of Mn atoms on the four interpenetrated sublattices in the fcc structures and the coupling through an RKKY interaction, also predicts that such a system will display a disorder induced phase transition at T = 0 and x = 7.5%. This is very close to the estimated value of x for which T c → 0. As occurs with previous experimental estimations for Co/CoO bilayers, the low value of β is in agreement with the predicted theoretical exponents for short ranged zero-temperature 3d models with metastable dynamics. However, the value of βδ falls out of the error bars of the numerical estimations. It is worth to note that Berger et al. claimed that their system, in contrast to our's, behaved as a 2d system. This is, however, in contradiction with the fact that for the 2d case, no disorder induced phase transition is theoretically expected [12].
In conclusion we have proven the existence of criticality in the magnetic hysteresis loops in Cu-Al-Mn. The non-equilibrium critical line has been determined in the (H, T, x) diagram. Both temperature and composition are the variables that allow to tune the critical amount of disorder. The coexistence of ferro and antiferromagnetic components in the system is at the origin of the observed behaviour.