Observation of unusual critical region behavior in the magnetic susceptibility of EuSe

The Europium Chalcogenides (EuCh), specifically EuO, EuS, EuSe and EuTe, have been regarded as prototype examples of Heisenberg exchange coupled magnetic systems. They have a simple FCC NaCl crystal structure and most observations of their magnetic properties appear to be well explained by a simple two-neighbor exchange model. The nearest-neighbor (NN) exchange constant J₁ is ferromagnetic (FM) while the next-nearest-neighbor (NNN) exchange constant J₂ is antiferromagnetic (AFM) [except possibly in EuO, where J₂ may also be FM]. EuO and EuS are ferromagnets, while EuTe is an antiferromagnet. EuSe appears to be on the cusp of being uncertain which one to be, and as a result displays a variety of strange and unusual magnetic behaviors (see for example Ref. 1). The basic reason for this has been attributed to EuSe having J₁ and J₂ that nearly cancel (J₂ ∼-J₁).

The magnetic structures in the EuCh can all be visualized as a stacking of ferromagnetically aligned sheets of spins lying in the (111) planes. A given (111) plane contains 6 NN spins, while each of the two surrounding (111) planes contain 3 NN spins and 3 NNN spins. Thus, if J₂=-J₁, in the mean-field approximation, a given plane is magnetically de-coupled from its two neighboring (111) planes (because of cancellations of the exchange fields) and thus becomes a stacking of non-interacting two-dimensional (2D) magnets. Presumably EuSe is very close to the point where J₂=-J₁, and therefore it should not be surprising to find that EuSe displays 2D behavior.^1,2 As is well known the Mermin-Wagner theorem (MW)³ shows that exchange-coupled 2D magnets cannot order above T=0 in the absence of anisotropy forces [although empirically 2D magnetic systems with high T_c’s do exist, despite very small anisotropy couplings]. Conventional spin-wave theory predicted that the ordering T_c should be zero, and the magnon dispersion in the <111> directions flat (as observed in quasi-2D systems) when J₂ becomes equal to -J₁.⁴ EuSe is presumably close to such a 2D-like "instability point." MW has been shown to apply in 3D magnets in special circumstances (high T_c superconducting parent compounds, where the magnetism becomes quasi-2D).⁵ We have also been able to apply this approach to show that in the EuCh exchange model, MW gives the rigorous result that no magnetic ordering can occur at the point where J₂ becomes equal to -J₁, thus supporting the idea that EuSe is 2D, or nearly so. The proof is based on the fact that when J₂=-J₁, the leading term in the dispersion relation goes as k⁴, and not k², analogously to Ref. 5, and so the "3D MW result" follows.⁵ 

The near decoupling of the EuSe (111) sheets of spins is presumably responsible for the sheets becoming easily rearranged into the three different magnetic structures that have been found in EuSe,¹ the coexistence of these magnetic structures over a range of temperatures (a very unusual occurrence), and the conversion of these structures into each other by weak influences (e.g., small fields, small pressures, etc).¹ In this report we describe two additional strange and unusual behaviors of EuSe. Above T_c the susceptibility, χ, of EuSe is observed to exhibit an unusual critical exponent γ, with magnetic ordering predicted not to occur till ∼0 K. On the two sides of the actual T_c, we observed a very strange and definitely unusual relaxation effect, as will be described below.

The susceptibility of EuSe was made in a Quantum Design (QD) Superconducting Quantum Interference Device (SQUID) magnetometer on a 50 mg powdered sample, from the third of three preparations (believed to be the best, and having a paramagnetic Curie temperature θ of 9.4K). The results were compared and were in agreement with independent χ measurements made previously during sample characterization at LSU,⁶ using material from the same sample preparation. The χ of a different sample of EuSe placed in a pressure cell was measured at pressures of 0, 2.5, 5, and 10 kbar in a QD SQUID, and gave somewhat different results at 0 kbar.⁷ The 10 kbar measurements, however, allowed the determination of the exponent γ for the FM state of EuSe (EuSe becomes FM above ∼5 kbar),² giving a γ of ∼1.40±0.05.

Fig. 1a shows a typical susceptibility (χ) measurement using a constant (DC) field of 5 G. The critical behavior above T_c is expected to follow the equation χ = $χ2Tc$(T/T_c-1)^-γ. Fitting the data is not a straightforward process, however. First, the temperature range in which this equation is valid was determined empirically, using a graphical method. Rewriting the equation in the form (1/χ)^(1/γ) = A(T-T_c), one can use EXCEL to graph the y-variable using the data values of χ and a sequence of guesses for γ, until a straight line is obtained for a region of temperatures above T_c, showing the equation is valid for that γ. In this way it was established that for γ=2, a straight line was obtained in the temperature range 5.0K to ∼9K. Fig. 1b shows the result for one of our data sets. At this point a curve-fitting program (Mathematica, in our case) was used to obtain fits of the data to the critical region equation, for the temperature range 5K to ∼9K. For several different data runs, at various magnetic fields, the results gave values of γ such as 2.05±0.05, 1.95±0.05. Some deviated from 2.0 by larger amounts, but all values centered on γ=2. At this point we were forced to assume that the unusual exponent of 2 is what the data is mandating. Assuming γ=2, one can then make linear fits, of the type shown in Fig. 1b, and found that the fitted T_c values [the point at which the system expects to order magnetically in going from the paramagnetic region (as χ→∞)] were all clustered near 0 K (+0.17K, +0.02K, −0.08K, etc.) for different temperature runs, and at various values of applied field.

Thus, it appears that the data show that EuSe has a very unusual critical exponent, not predicted by any current theory. Additionally, we interpret T_c ∼ 0K to mean the system in the paramagnetic state "thinks" it should not order above 0 K, but something "triggers" it to do so at 4.7K. The notion of "triggering" magnetic ordering in 3D for quasi-2D magnets was advanced at the outset of the discovery of quasi-2D magnets,⁸ but no mechanism has so far been advanced to support this concept. To see if other 2D magnets displayed similar behavior, we measured χ in the well-studied quasi-2D system CrCl₃. Surprisingly, a γ of 2.0 was also found in this case. However, the predicted paramagnetic T_c came out as 10.3K, whereas the actual (measured) T_c is near 16K, still constituting a sizable difference.⁹ It will be interesting to check other 2D magnets (especially those with lengthy organic spacers)¹⁰ to see if similar behaviors obtain there as well. We believe that our observations give credence to the "triggering" concept. Above T_c, the spins are coupled primarily via Heisenberg exchange. However, we suggest that at some point (here being 4.7K), the spin system goes into some kind of a strongly correlated spin state, with the spin interactions no longer being merely the sum of independent J S_i.S_j Heisenberg-interaction terms, but acquiring some sort of overlying collective exchange interaction modification.

As a caveat to our above-described results regarding γ and the associated critical-equation-predicted T_c we note the following. EuSe samples are known to show significant differences in magnetic behaviors from minute and unidentifiable changes in sample, or difference in preparation.¹¹ This was the case here, as well. The first sample of our preparation, and the sample used at ISIS,⁷ both gave different fitted γ′s (∼1.6 to ∼1.7) and correspondingly different fitted T_c′s (0.8K-1.5K). Our own sample showed a similar change, when part of it was exposed to air for ∼3 days. The XRD spectra showed no discernible changes.

The other unusual, and indeed very strange, critical behavior of EuSe, was discovered by accident, and involves a novel relaxation phenomenon. When the sample of EuSe is quickly lowered (in a fixed small field) from a fixed higher temperature (say 20K, but higher and lower temperatures were also tried) to just below the measured T_c of ∼4.7K, the initial measured magnetic moment is higher than normal, and on repeating the measurement immediately thereafter (which typically takes ∼90 sec in the SQUID) gave lower and lower values of the moment, until a plateau was reached. When the temperature was lowered quickly from say 20K to just above T_c, the effect was inverted, i.e., the initial moment measured was smaller than the final plateau value. An overview of this phenomenon is shown in Fig. 2a. Representative "relaxation curves" are shown in more detail in Fig. 2b for two neighboring temperatures below T_c. Our rough estimate of the relaxation time constant is 5 minutes or less; however, fitting using an exponential curve did not give better fits than our rough estimates, since the measuring window (for one measurement) of the SQUID is about 100 sec and therefore distorts the exponential curve substantially.

The variation of the magnitudes of the susceptibility deviations Δχ as a function of temperature is shown in Fig. 2c, where Δχ represents the difference between the before-relaxation susceptibility χ_initial and the after-relaxation χ_plateau. We see that the effect is not symmetrical, being detectable as much as to ∼0.3K below, and ∼1.8K above T_c. A comparison of our "normal" measurements, where the susceptibility curve is measured by a sequence of slow, small changes in temperature (as in Fig. 1a), showed that the "after relaxation" values coincide exactly with such "normal" measurements.

We do not yet have any good physical explanation for the observed relaxation effect. Measurements on CrCl₃ near T_c show no such effect. Using other materials, as well as CrCl₃, no time dependence (or any abnormality) was observed in measuring moments near 4.7K, so it is definitely not an instrumental problem. The EuSe measurements were repeated using zero-field cooling (ZFC). The spikes below T_c appeared similar, but fell off at lower temperatures a little bit faster than shown in Fig. 2a.

Assuming the observed relaxation effect is indeed physical, we think we can make at least a tentative interpretation, as follows. In the region just above, and just below T_c, the spin system appears to thermally decouple from the lattice, with one exception. When the temperature is swept down quickly from 20K, to T_c=4.68K (± ∼0.01K), the spin system and the lattice appear to stay at the same temperature. But when the temperature is swept down through and below T_c, the moments measured appear to show that for the approximate region down to ∼0.10K below T_c (measured every 0.01K), the initially measured moments stay at nearly the T_c value, as if after passing through T_c, the spin system temperature stays completely decoupled from the lattice, and then starts coming to equilibrium with an astonishingly long time constant (as much as ∼5 min). By contrast, when the temperature is lowered quickly from 20K to just above T_c, the spin system starts lagging the lattice temperature (starting from about 1.8K above T_c), appearing to stay at a higher temperature than the lattice, again relaxing thereafter, and coming to equilibrium, though with a smaller time constant. We stress again that measurements of other materials in our SQUID show an immediate response of the measured magnetic moments to the lattice temperature (which is what the SQUID thermometer measures) throughout the temperature region where the EuSe relaxation effect was observed. An additional - sideline comment - is that the measurements depicted in Fig. 2a allow us to pin down the measured value of T_c of our EuSe sample much better than normal, i.e., to within ∼0.005K (T_c∼4.685K). We have no explanation yet for these observed relaxation effects, or why the presumed spin-lattice temperature decoupling should occur, but have no reason to doubt that the effect is physical, since it was consistently reproducible, and so intimately tied to the EuSe T_c.

We have made various precision susceptibility measurements in EuSe and have discovered two new and unusual behaviors (an unusual critical exponent and an unusual relaxation effect). The paramagnetic critical region exponent γ has a value of 2, an exponent not predicted by current phase-transition theories for either 3D or 2D Heisenberg systems. Additionally, these data predict that magnetic system should stay in the paramagnetic state until T_c=0K, whereas in fact EuSe appears to be "triggered" into an ordered state at 4.7K. EuSe in a FM state (under 10 kbar pressure) shows a γ of ∼1.4 and a fitted T_c of ∼4.3K (measured T_c∼6.0K). The change in γ as a function of pressure may represent a violation of the Universality Hypothesis of critical theory, which predicts that γ should remain unchanged (unless the dimensionality of the magnetic interactions has changed). The second unusual behavior found in the susceptibility of EuSe consists of a relaxation effect with an astonishingly large time constant (of up to 5 min), both above and below T_c. The data appear to show that the spin system and the lattice can get thermally decoupled in EuSe near T_c for as yet unknown reason.