Issue 1, August 2003

Physical Sciences & Mathematics
Magnetic Properties of Single-Grain Icosahedral Rare-Earth-Mg-Cd Quasicrystals Prepared from a Ternary Melt

Tony Huie
Stanford University - Stanford, California
Advisor: Ian R. Fisher, Ph.D.
Department of Applied Physics, Stanford University - Stanford, California
Graduate Mentor: Suchitra Sebastian
Department of Applied Physics, Stanford University - Stanford, California

Abstract

In this paper we describe a method for growing high-purity single-grain R-Mg-Cd quasicrystals. For Rare Earth (R) elements Dy, Tb, and Gd, we report dc magnetization data taken from single quasicrystals. These quasicrystals demonstrate a temperature dependence that obeys the Curie-Weiss law at elevated temperatures, T > 50 K, and a freezing behavior similar to canonical spin-glasses at low temperatures. In particular, the quasicrystal Gd-Mg-Cd displays an unusual correlation between its freezing temperature and the strength of magnetic interactions in comparison to Tb and Dy quasicrystals.

Introduction

The arrangement of atoms in a conventional crystal is both ordered and periodic. In other words, all the atomic positions within a crystal can be described by an integer set of translation of a unit cell. Symmetry and the requirement that all of space be tiled limits the number of allowed crystal lattices.

A quasicrystal’s ordered aperiodicity can be understood from drawing an analogy to the well-known Fibonacci sequence. Any term of the Fibonacci sequence can be deduced by simply summing the two terms previous, and in this way one is able to construct a sequence of terms that has no duplicated core or “unit” of numbers, in contrast to a simple arithmetic sequence. Thus, although the Fibonacci sequence is not periodic, its systematic construction is completely ordered. Quasicrystals, then, are essentially three-dimensional structures of a “Fibonacci-like” sequence of atoms: an ordered crystalline solid with no single, repeatable unit cell.

Because of their unique spatial arrangement of atoms, the behavior of local magnetic moments in this quasiperiodic structure has been a topic of great interest. To study this experimentally, it is necessary to use quasicrystals with well-localized magnetic moments. Up until only recently, the R-Mg-Zn (R = Rare Earth) quasicrystals were the only class to demonstrate such properties (Niikura, et al. 1994). The discovery of quasicrystals with well-localized moments in the R-Mg-Cd alloys has allowed comparison of magnetic properties in quasicrystalline solids (Guo, et al. 2000). The results of early experimentation on polygrain crystalline samples of R-Mg-Cd can be briefly summarized as follows: At elevated temperatures, the susceptibility of R-Mg-Cd polygrain samples obeys the Curie-Weiss law. This law describes how free magnetic moments in a material respond to a magnetic field as a function of temperature. Estimated effective magnetic moments of those samples corresponded to those of free R³⁺ ions, confirming the well-localized moments originating from the 4f electrons in the R atoms. However, at lower temperatures the polygrain samples appeared to show an unconventional two-step freezing process (Sato, et al. 2000). Similarly, previous measurements of R-Mg-Zn quasicrystals also showed a spin glass state but with only one freezing transition, in line with canonical spin glasses. A spin glass is a magnetic state in which the moments are frozen in a random configuration with each other, as opposed to magnetic states that exhibit long-range order such as ferromagnets or antiferromagnets. A two-step freezing process would be unconventional, and we were motivated to investigate this using high-quality samples.

In this paper, we demonstrate a technique to effectively produce relatively large, single-grain R-Mg-Cd quasicrystals and present magnetization data taken from these high-purity samples. By using these single-grain samples, we are able to measure the intrinsic properties of this material while eliminating the sources of impurities potentially present in polycrystalline samples. In this way, we were able to show that single-grain samples exhibit canonical spin-glass behavior characterized by a single freezing transition. This work clarifies our understanding of magnetism in quasicrystals.

Sample Preparation

High-purity samples were prepared using a self-flux growth technique. This procedure allows the growth of single-grain quasicrystals via the slow cooling of a ternary melt that intersects the primary solidification surface of the icosahedral phase, providing a distinct advantage to traditional casting techniques, which typically yield lower-purity, polygrain samples. The flux growth technique is similar to the growth of salt crystals out of water. The lower melting point flux is used to grow the high melting point crystals. In this case, we used a "self-flux" so as not to introduce foreign elements. Experimentation is necessary to determine the best composition of the ternary melt.

Figure 1. Ternary phase diagram for R-Mg-Cd. Ranges of attempted melt compositions were 0.6-0.7 for Cd, 0.225-0.375 for Mg, and 0.025 for R, and are shaded on the diagram. R0.025Mg0.275Cd0.7 produced consistent results and is the melt composition of samples used in this study. The composition of the resulting quasicrystals is R0.1Mg0.4Cd0.5 and is marked in the diagram with a star.

Purities of the starting elements were 99.995%, 99.9999%, and 99.98% for R, Cd, and Mg, respectively. The percentages of Cd and Mg were varied from 60-70% and 22.5-37.5% respectively, while the proportion of the Rare Earth element was held at 2.5%. The R_2.5Mg_27.5Cd₇₀ melt composition produced consistent icosahedral morphologies; thus, our magnetic property analysis is carried out primarily from quasicrystals grown from a melt of this starting composition. The ternary phase diagram shown in Figure 1 plots the experimented compositions as well as the actual sample composition marked by a star. The initial composition of the ternary melt used for flux growth differs from the desired R-Mg-Cd compound, which has a composition of approximately R₁₀Mg₄₀Cd₅₀ as measured by energy-dispersive X-ray spectroscopy.The starting elements were sealed in Tantalum (Ta) tubes, which were then sealed in quartz tubing and placed inside high-temperature bricks. The melt was heated up to 700°C and slowly cooled to 400°C to allow sufficient crystal formation. It was then taken out at a temperature of approximately 400°C and the remaining flux was decanted. The temperature profile is given in Figure 2 along with a schematic of the growth set-up. The single grain R-Mg-Cd quasicrystals most frequently demonstrated a rhombic triacontahedral morphology. The structure of samples used in this study was verified by transmission electron microscopy.

Figure 2. Temperature profile for crystal growths. The melt was heated to 700° C, cooled to 400° C, and decanted. Inset: schematic of growth set-up (Ta = Tantalum).

Experimental Methods

DC magnetic measurements were taken using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design, MPMS-XL). This device works for moving the magnetic material through a set of sensing coils. The temperature dependence of dc magnetization was measured for temperatures from 2 K to 300 K under several externally applied magnetic fields. Two sets of data were taken for each sample: a Zero Field Cool (ZFC) measurement, for which the sample was cooled with no externally applied field, and a Field Cool (FC) measurement, for which the sample was cooled under external magnetic fields. Under the ZFC and FC conditions, the magnetization was then measured for increasing T. Low-Temperature measurements were made in a field of 100 Oe, and high-temperature measurements were made using a 1000 Oe field. The results are presented using magnetic susceptibility, χ, which in the low-field limits, is defined as M / H (M = magnetization, H = magnetic field).

Results

Figure 3. Photograph of a Tb-Mg-Cd quasicrystal over a 1 mm scale. This sample’s rhombic triacontahedral morphology is clearly illustrated, consistent with an icosahedral symmetry. Note the axis of five-fold symmetry, forbidden for conventional crystals. Note also the small droplets of undecanted flux on the facets of the sample.

Figure 3 shows a photograph of a Tb-Mg-Cd quasicrystal taken against a 1 mm scale. Figures 4

and 5 show the magnetic susceptibility, χ and χ^-1, for R = Dy, Tb, and Gd, shown on a temperature scale of 0 – 10 K for χ and 0 – 300 K for the inverse susceptibility plot. The inverse magnetic susceptibility shows a linear relationship above about 50 K that extends to high temperatures, as expected from the Curie-Weiss Law, evidencing the presence of well-localized magnetic moments. Between 50 K and 300 K, the data can be well fitted using the conventional formula:

, (1)

where χ₀ , N_A, μ_eff, k_B, and θ are a temperature-independent term, the Avogadro number, the effective moment, the Boltzmann factor, and the Weiss temperature, respectively. The estimated μ_eff and θ are listed in Table 1 along with the free R³⁺ ion moments calculated from μ_R³⁺ = , where g is the Landé g-factor and J is the total angular momentum. Noting that the effective moments μ_eff are relatively close to the free ionic moments μ_R³⁺, it can be concluded that the R atoms are trivalent in the R-Mg-Cd quasicrystalline system, as expected.

Table 1. Effective moments μeff and the Weiss temperature* *Calculated from the Curie-Weiss fit along with the moments of the free Rare Earth ions μR3+. †The estimated freezing temperatures Tf taken from the magnetic susceptibility data (H = 100 Oe) with estimated uncertainty are also listed.

At lower temperatures, ZFC and FC magnetization data were measured under an externally applied field of 100 Oe to check for spin-glass-like freezing, because this freezing was commonly observed in previously examined R-Mg-Zn quasicrystals (Niikura, et al. 1994). All three data sets show a clear irreversibility below a well-defined freezing temperature. This behavior is consistent with the freezing commonly observed in canonical spin-glasses (Mydosh 1993). The broad transition temperature in the Tb magnetization data is likely extrinsic in origin, and may be the result of the tested sample’s small surface-area-to-volume ratio, intensifying any effects of surface impurities on the magnetic susceptibility measurements.

Figure 6(a) shows the Weiss temperature plotted against the de Gennes factor

dG = (g-1)2J(J+1), which is a measure of the strength of magnetic interactions (Blundell 2001). The figure demonstrates that the Weiss temperature can be well scaled by the dG factor, where greater Weiss temperatures correspond to stronger magnetic interactions. By contrast, the freezing temperatures, shown in Figure 6(b), demonstrate an unusual trend inconsistent with the Weiss temperature scaling. The trend of increasing freezing temperature versus dG is followed for both the Dy and Tb samples. However, the Gd sample, having the strongest magnetic coupling of the surveyed rare earth elements, exhibits a freezing temperature that sharply deviates from the magnetic interaction strength correlation demonstrated in the Weiss temperature plot.

Figure 6. The deviation of the Gd-Mg-Cd freezing behavior from Weiss Temperature trends is shown. (a) The magnitude of the Weiss Temperature versus the de Gennes factor dG = (g-1)2J(J+1) for R-Mg-Cd quasicrystals. Error margins are result of Curie-Weiss fit for T > 50 K. (b) The observed freezing temperature Tf versus dG.

Discussion

A spin glass is a material that has random coupling between magnetic moments, resulting in a randomly frozen pattern at low temperatures. There are several effects that can lead to spin glass behavior. In canonical spin glasses, such as Copper (Cu) with 1% of the sites substituted with magnetic Manganese (Mn), it is the random distance between the Mn atoms that leads to this effect. At low temperatures, the moments freeze in a random configuration with no long-range order. In the case of the R-Mg-Cd quasicrystals, the structure is not random, so it appears to be the non-periodicity that ultimately causes the R magnetic moments to freeze into a spin glass state.

The spin freezing temperature, T_f, can be readily found by examining the magnetic susceptibility of a material. Above T_f, thermal fluctuations dominate the magnetic properties, and the moments follow the usual Curie-Weiss temperature dependence. Below T_f, the moments cannot rotate so easily with respect to each other. There are therefore large differences between ZFC data, for which the moments are frozen with no directional preference, and FC data, for which the moments are frozen while the applied field partially aligns them. Typically, if a material has a strong coupling between the moments, it will freeze at a higher temperature. However, the Gd-Mg-Cd quasicrystals show a peculiar deviation in the trend; the Gd sample’s coupling interactions are stronger than the Tb sample, but exhibit a lower freezing temperature. This break in the trend is likely the result of Gd’s half-filled 4f electron shell, and is described below.

At high temperatures, thermal energy leads to the Curie-Weiss law for magnetic susceptibility. However, at low temperatures the effects of thermal energy get progressively weaker compared to magnetic interactions between the moments. In R-Mg-Cd quasicrystals, the moments freeze in a random configuration due to the non-periodic atomic structure. However, the local environment also plays a role in this process through what is known as the Crystal Electric Field (CEF). The CEF can limit the number of directions in which each magnetic moment can point, thus substantially increasing the effective degree of magnetic frustration. This appears to be the case for R = Dy and Tb, which have very non-spherical electron distributions. However, R = Gd is an exception. This ion has an exactly half-filled 4f shell, which is therefore spherically symmetric. In effect, the magnetic moment of Gd can point in any direction irrespective of its environment. Since Gd is insensitive to the local CEF environment, Gd-Mg-Cd freezes at a lower temperature than either Dy-Mg-Cd or Tb-Mg-Cd, in spite of its stronger magnetic coupling.

It is significant that our measurements show no sign of a two-step freezing process, in contrast to previously published data for polygrain crystalline samples (Sato, et al. 2000). We suspect that the two-step freezing previously reported may be the result of contaminating impurities with a different freezing temperature than the desired sample. The polycrystalline samples used in the previous study are prone to contaminating crystalline phases that may cause the sample to exhibit artificial properties. We note that the width of formation for R-Mg-Cd quasicrystals may be sufficiently large so that the two-step freezing may be a property of R-Mg-Cd quasicrystals of slightly different compositions, though this seems unlikely to us. Our single-phase, single-grain samples only show one freezing transition consistent with usual spin-glass behavior.

Conclusions

We have demonstrated that large, single grain R-Mg-Cd quasicrystals can be readily grown using a ternary melt, and are suitable for magnetic analysis. The magnetic properties of these single R-Mg-Cd quasicrystals can be well represented by the Curie-Weiss Law at higher temperatures, and exhibit a single-step freezing process similar to canonical spin-glasses. However, the freezing temperature does not scale with the de Gennes factor for Gd, demonstrating the importance of local environment on spin freezing phenomenon. This work has clarified our general understanding of magnetism in quasicrystalline materials and shown that single crystals provide the best measure of intrinsic properties.

Acknowledgements

The author would like to thank the Fisher Group: S. Sebastian, Y. Matsushita, T. Holme, N. Ru, K. Shin, and D. DaMann for stimulating discussions. A special thanks to Prof. Ian Fisher for granting me the tremendous opportunity to conduct research so early in my academic career and for his continual guidance and mentorship. Support for this work has been provided by the National Science Foundation and the Stanford Vice Provost for Undergraduate Education (VPUE) through the Stanford Physics Summer Research Program.

Figures 4 and 5

Figure 4. Magnetization curves of R-Mg-Cd measured under H = 100 Oe showing both FC and ZFC behavior. The freezing temperature, Tf , is indicated by an arrow. (a) Dy-Mg-Cd. (b) Tb-Mg-Cd. (c) Gd-Mg-Cd.

 

Figure 5. Inverse magnetic susceptibilities (H/M) of magnetic R-Mg-Cd quasicrystals measured under H = 1000 Oe. Solid lines show Curie-Weiss fit, with parameters as given in Table 1.

 

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References

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Journal of Young Investigators. 2003. Volume Eight.
Copyright © 2003 by Tony Huie and JYI. All rights reserved.