Magnetism of Tsai-Type Quasicrystal Approximants

Abstract

Magnetism of Tsai-type 1/1 approximants has been extensively investigated for the last decade. The experimental results on Tsai-type approximants are reviewed and the magnetic behaviors are classified and discussed in terms of the average electron-per-atom (e/a) ratio. The paramagnetic Curie-Weiss temperature Θ_p is found to oscillate in a universal manner as a function of the e/a ratio, and the magnetic order systematically changes with the variation in Θ_p. In addition, recent work shows that the 2/1 approximants also follow the trend observed for the 1/1 approximants, which implies that the magnetic phase diagram of the 1/1 approximants can be applicable to higher-order approximants including quasicrystals that are yet to be discovered.

1. Introduction

Since the discovery of quasicrystals (QCs) that have rotational symmetries forbidden in periodic crystals,¹⁾ particular attention has been paid to their unique physical properties intrinsic to the quasiperiodicity. After the discovery of Al-based stable QCs,^2,3) anomalously high electrical resistivity was identified and even an insulating behavior was observed for the Al–Pd–Re QC.^4,5) Moreover, in 2012, quantum critical phenomenon was discovered in the Au–Al–Yb QC and highly robust nature of the quantum criticality against pressure in the QC was demonstrated,⁶⁾ whereas the Au–Al–Yb 1/1 approximant (AP) in the nearby composition is found to be off the quantum critical point, showing that the QC is located or locked at the quantum critical point, which suggests that the quantum criticality can be an intrinsic property to the QC, or, to the quasiperiodicity. As for the magnetism, theoretical works have shown that an antiferromagnetic (AF) order is possible in QCs.⁷⁾ However, magnetic order has not been discovered in real quasiperiodic lattices yet and is still an on-going fundamental issue in solid state physics. Meanwhile, magnetic orders have been found in a number of Tsai-type APs since the first discovery of an AF transition in the Cd₆Tb AP in 2010⁸⁾ and the condition for the realization of magnetic orders in QCs has now been getting unraveled during the last decade by both experimental and theoretical^9,10) approaches. The aim of the present paper is to review and discuss the current status of the magnetism research in R-containing Tsai-type APs in order to stimulate the search for unprecedented magnetic QCs.

The materials reviewed in this paper are rare-earth (R) containing Tsai-type APs, only in which magnetic orders have been reported to date. Most extensively investigated Tsai-type APs are 1/1 APs simply because of their abundancy. Figure 1 shows the structure of Tsai-Type 1/1 and 2/1 APs. The Tsai-type 1/1 AP can be regarded as a bcc packing of Tsai-type icosahedral clusters. Both the Tsai-type 1/1 and 2/1 APs can be described as a packing of rhombic triacontahedron (RTH) clusters. The RTH cluster consists of five concentric shells, from the center outwards: a tetrahedron of 4 atoms, a dodecahedron of 20 atoms, an icosahedron of 12 R atoms, an icosidodecahedron of 30 atoms, and an RTH of 92 atoms. We note that there exist some variations at the center of the RTH cluster; either one atom or four atoms occupy the cluster center depending on the system or the composition, the reason of which itself is subject to further studies. Concerning the position of the R atoms, R atoms exclusively occupy the second icosahedron shell and other shells are occupied exclusively by other metallic elements. To summarize, the Tsai-type 1/1 APs can be basically viewed as a bcc network of spin icosahedra, providing a unique magnetic system that has not been investigated before.

Fig. 1

Two building blocks of Tsai-type 1/1 and 2/1 approximants: (a) Tsai-type rhombic triacontahedron (RTH) cluster and (b) acute rhombohedron (AR). Red and blue balls represent R sites. Arrangements of R atoms in (c) 1/1 and (d) 2/1 approximants.

2. General Trend of the Magnetism of Tsai-Type 1/1 Approximants

Table 1 shows all the Tsai-type QCs and APs reported to date and Table 2 lists the compositions of all the R-containing Tsai-type APs investigated so far,^11–83) together with the information on their magnetic properties such as the ferromagnetic (FM) transition temperature T_c, the AF transition temperature T_N, the freezing temperature T_f, the paramagnetic Curie-Weiss temperature Θ_p and the effective magnetic moment μ_eff. For all the APs, the values of μ_eff are close to the theoretical values expected from free R³⁺ ions. We note that for Tsai-type 1/1 APs, all the R atoms are under the same environment since the R atoms occupy a single crystallographic site as mentioned above. The good agreement of μ_eff with the theoretical values of the free ions means that the 4f states can be regarded to be under central potential to a first approximation, i.e., under weak crystal electric field. Thus, R³⁺ spins are well localized at the vertices of the second icosahedron shells. On the other hand, Θ_p represents the net magnetic interaction on each spin which is induced by the external magnetic field. If one assumes that interactions between R³⁺ spins are mediated by conduction electrons, i.e., via the RKKY interaction, as is often the case for R-containing metallic compounds, Θ_p is expected to be proportional to de Gennes factor dG. In such a case, Θ_p/dG is independent on the system provided that the other factors such as the Fermi wavevector k_F, etc., are the same. Figure 2(a) shows Θ_p/dG as a function of the e/a ratio for the R (R = Eu, Gd, Tb)-containing Tsai-type APs reported to date. The number of the valence electrons assumed for each element is listed in Table 3. Two significant features can be extracted from Fig. 2(a). One is that Θ_p/dG is rather strongly dependent on the e/a ratio, which means that k_F is a rapidly varying function of the e/a ratio. Second is that the plot points toward the existence of a universal curve irrespective of systems or atomic species, which indicates that Θ_p is mainly determined by the e/a ratio, and the role of the chemical species in the magnetism is rather limited. The existence of such a universal behavior also justifies the simple assumption of the valence electron for each element (Table 3). Here, one exception to the universal trend is noted; the magnitudes of Θ_p of the Ag-based APs are appreciably larger than those of the other APs having similar e/a values, the reason of which is not clear at the moment.

Table 1 Rare-earth-containing Tsai-type approximants.

Table 2 Magnetic properties of the rare-earth containing Tsai-type approximants.

Fig. 2

(a) The paramagnetic Curie-Weiss temperature normalized by the de Gennes factor Θ_p/dG as a function of the electron-per-atom (e/a) ratio for the R(R = Eu,Gd,Tb)-containing Tsai-type APs reported to date. (b) The magnetic order at the lowest temperatures as a function of the e/a ratio.

Table 3 The number of the valence electrons assumed for each element in the estimation of the e/a ratio.

3. Magnetic Orders and Their Dependence on the e/a Ratio

Figure 2(b) shows the magnetic order observed at the lowest temperatures, where the magnetic order can be grouped into three distinctive regions according to the e/a ratio; AF, FM and spin-glass-like regions. This clearly means that the e/a ratio is a good controlling parameter of the magnetism of the Tsai-type 1/1 APs. Spin-glass-like behaviors are observed for e/a = 1.84∼2.16, when Θ_p is large negative. FM orders are observed for e/a = 1.58∼1.84, when Θ_p is large positive. AF orders are observed for e/a = 1.54∼1.70, when Θ_p is weakly positive or weakly negative. For Tsai-type approximants and quasicrystals, the magnetic interaction is considered to be mediated via conduction electrons thorough the RKKY interaction. In this scheme, the change in the e/a ratio will result in a change in the Fermi wavevector k_F and, hence, in the oscillation of the magnetic interaction between each pair of spins from minus to positive or positive to minus. Since the Weiss temperature Θ_p is regarded to be proportional to the sum of all the interspin interactions, the observed oscillation of Θ_p is then understood as a result of continuous change in k_F through the variation in the e/a ratio. Here, we notice substantial overlap of the AF and FM regions in Fig. 2(b). The main reason for such overlap is that the AF region of the Tb system is considerably wider than those of the Gd and Eu systems, which implies that the AF state is stabilized under uniaxial anisotropy. On this point, simulation on a classical spin icosahedron, the method of which will be described later, also shows that the AF/FM boundary moves toward the FM region with the introduction of uniaxial anisotropy to a Heisenberg icosahedron, in qualitative agreement with the experiment.

For the ferromagnetism, the FM order of the 1/1 Au–Si–Tb AP has been recently solved by Hiroto et al.,⁷³⁾ the magnetic structure of which is shown in Fig. 3. The spin configuration is non-coplaner and is understood by considering that the easy axis lies inside the mirror plane and is nearly perpendicular to a pseudo-fivefold axis. Figure 4 shows the AF order of the 1/1 Au–Al–Tb AP recently solved by Sato et al.⁸⁴⁾ The spin configuration of the AF order is also non-coplaner and is also understood by considering that the easy axis direction is similar to that of the FM phase. Thus, for both the FM and AF phases, there exists strong uniaxial anisotropy for a Tb³⁺ spin and, therefore, the Tb³⁺ spins can be regarded as Ising spins. Energy calculation of an Ising icosahedron taking into accounts the nearest neighbor (J₁) and second nearest neighbor interactions (J₂) with the Hamiltonian $H = - \sum\nolimits_{\langle ij\rangle }J_{ij} \vec{R}_{i} \cdot \overrightarrow{R_{j}}$ where $\vec{R}_{i}$ and $\overrightarrow{R_{j}}$ the positions of i-th and j-th spins, respectively, were performed over the 2¹² possible states by Sato et al.,⁸⁴⁾ which shows that the observed AF structure is stabilized under the condition that J₂ is positive and J₁ < J₂/2. Interestingly, J₁ can be both positive and negative for the AF state, implying that the second nearest spin interaction plays a dominant role in the AF state. The calculation is in good agreement with the experimental observation that the AF phase is obtained when Θ_p is positive. According to the above calculation, the ground state becomes highly degenerate when J₂ is large negative, or, for large negative Θ_p. The occurrence of a spin-glass state indicates the existence of strong frustration for this large negative Θ_p region. Here, one exception is noted for the spin-glass region; 1/1 Cd₆R APs exhibit successive AF transitions instead of spin-glass-like freezing.^8,28) Considering that there exists inherent chemical disorder at the non-R sites for ternary systems, the occurrence of spin-glass-freezing may be attributed to the interplay between the high degeneracy and chemical disorder. Clarification of the origin of the AF behaviors in the 1/1 Cd₆R APs is certainly a future issue.

Fig. 3

Magnetic structure of the FM 1/1 Au–Si–Tb AP. The spin configuration on the icosahedral cluster at the body center position is illustrated and this unit forms a bcc lattice.⁷³⁾

Fig. 4

(a), (b) Magnetic structure of AF 1/1 Au–Al–Tb AP. For each spin, the spin at the opposite vertex of the icosahedron is antiparallel, which explains the occurrence of the AF order. The spin structures at the origin and at the body center (BC) are related by the time-reversal operation, breaking the chemical bcc symmetry.⁸⁴⁾

In order to further understand the experimental trend for APs (Figs. 2(a) and 2(b)), we performed numerical calculation of the total energy and the sum of interactions for an isolated icosahedron cluster based on Ising spin model. Here, the easy axis is chosen to be perpendicular to the fivefold axis of the cluster and also taken to be inside the mirror plane, in order to take into account the experimental observations mentioned before. The model Hamiltonian is described as below.   

\begin{equation*} \mathcal{H} = -J_{1}\sum_{\langle i,j \rangle_{1}} \boldsymbol{S}_{i} \cdot \boldsymbol{S}_{j} - J_{2}\sum_{\langle i,j \rangle_{2}} \boldsymbol{S}_{i} \cdot \boldsymbol{S}_{j} \end{equation*}

Here, J_n and ⟨i,j⟩_n (n = 1,2) represent the exchange energy and nth neighbor bond, respectively. The spins are placed on the vertices of the icosahedron: There are 30 bonds for 1st neighbor interaction and 30 bonds for 2nd neighbor interaction. The calculation was performed over 2¹² states with changing the value of α_J, where α_J is defined by J₁ = cos(α_J) and J₂ = sin(α_J). Both the sum of all interactions and the total energy of a single icosahedral cluster are displayed in Fig. 5. Note that the horizonal axis is taken to be −α_J, which ranges from −π/2 to π/2, for the sake of comparison with Fig. 2. For the AF1 and AF2 states, the summation of all spins become zero showing that an AF order is realized. We note that the AF1 state has the lowest energy among all the obtained states indicating extremely high stability. The FM state next to the AF1 state has the largest moment size $2(\sqrt{5} + 1)$ compared with the weak FM1 (moment size: $4\sqrt{ {1 + 2/\sqrt{5} }\mathstrut} $) and weak FM2 (moment size: $4\sqrt{ {1 - 2/\sqrt{5} }\mathstrut} $) states. The sum of all the interactions, 30(J₁ + J₂), corresponds to Θ_p in the mean field approximation. The boundary between the AF phase and FM phase exists in a positive Θ_p region, which is in agreement with the experimental observation (Fig. 2). Also, highly frustrated nature in the negative Θ_p region is inferred from the calculation, in striking contrast with the high stability of the AF1 states in the positive Θ_p region, which is also in agreement with the experimental observation.

Fig. 5

The −α_J dependence of the sum of interactions, 30(J₁ + J₂), and the total energy obtained by numerical calculation.

Now, we focus on the AF region and discuss the dependence of Θ_p and T_N on the Au concentration, x. Figure 6 shows χ-T curves of the 1/1 Au–Al–Tb APs with different Au concentrations, x. All the APs exhibit a sharp cusp at T_N ∼ 11.7 K without accompanying bifurcation between the ZFC and FC susceptibilities below T_N, indicating the occurrence of an AF transition for all the samples. Θ_p is found to increase from 4.2 to 11.6 K with decreasing x inside the AF region, clearly showing that the FM contribution is dominant and the FM contribution becomes larger with the increase of x. It is also clear that there exists some finite domain for the AF phase and the change in x leads to the change in the relative contributions of FM and AF interactions. The maximum in χ significantly increases with decreasing x, which is also attributed to the increase of the FM contribution with decreasing x. On the other hand, the Néel temperature T_N does not change appreciably with x. Since there exist both FM and AF interactions, the molecular field H_m can be expressed as a sum of the two contributions, i.e., H_m = (λ_FM + λ_AF)M, where λ_FM > 0 and λ_AF < 0. Then, Θ_p is proportional to (λ_FM + λ_AF) and its sign is determined by the majority of the two contributions. If one assumes a simple case that both the FM and AF interactions stabilize the AF phase, e.g., J₁ < 0 and J₂ > 0 in the above simple model, T_N should be proportional to (|λ_FM| + |λ_AF|). Thus, the constant T_N with x can be most simply understood by considering a situation that the sum of the absolute values, i.e., |λ_FM| and |λ_AF|, remains nearly constant with the change in x.

Fig. 6

Magnetic susceptibility of 1/1 Au–Al–Tb APs with different Au concentrations. All the samples exhibit a sharp cusp at T_N ∼ 11.7 K. The maximum of χ significantly increases with decreasing the Au concentration x.

4. Toward Magnetic Tsai-Type Quasicrystals

Recently, magnetic transitions have also been observed in Tsai-type 2/1 APs.⁶⁶⁾ The structures of both the 1/1 and 2/1 APs are shown in Fig. 1. The main structural difference between the 1/1 and 2/1 APs is that the 2/1 APs have an acute rhombohedron (AR) unit which is missing in the 1/1 APs, and because of this the 2/1 APs have all the building blocks necessary for the construction of the Tsai-type QC, thus enabling to extract the effect of the quasiperiodic order when long-range magnetic order is realized in Tsai-type QCs. Concerning the R position, the R atoms are situated not only at the vertices of the Tsai-type clusters but also located inside the AR unit in contrast with the 1/1 AP. The Θ_p/dG values of the 2/1 APs reported to date are also plotted in Figs. 2(a) and 2(b), which shows the 2/1 APs follow the trend obtained for the 1/1 APs. In addition, FM and AF phases are observed when Θ_p is large positive and weakly positive, respectively, as in the case of the 1/1 APs. Spin-glass-like freezing phenomenon occurs when Θ_p is large negative, also in agreement with the case of the 1/1 APs, indicating the occurrence of strong frustration in the strongly negative Θ_p region. Figure 7 shows comparison of AF transitions between the 1/1 and 2/1 APs having similar e/a values in the same Au–Ga–Eu system. The AF transition is found to survive in the higher-order AP with nearly the same e/a ratio, indicating that the magnetic phase diagram obtained for the 1/1 AP is also applicable to the 2/1 AP. Concerning the χ-T and M-H curves, there is not much difference between the 1/1 and 2/1 APs. Indeed, both the AF phases commonly exhibit a spin-flop at a low field, characteristic to AF orders. Since the Eu²⁺ spin is isotropic, the existence of the spin-flop for the Eu²⁺ systems clearly indicates that the ground state is nearly degenerate and spin reorientation is required between the nearly degenerate states. Determination of the AF order in the 2/1 AP is certainly a future issue.

Fig. 7

Temperature dependences of ZFC and FC magnetic susceptibilities for (a) 1/1 Au₆₆Ga₂₀Eu₁₄ and (b) 2/1 Au₆₅Ga_20.5Eu_14.5 below 30 K. A sharp cusp due to an AF transition is observed at T_N = 7.0 and 8.5 K, respectively.⁶⁶⁾

Very recently, it has been found that isovalent substitution can stabilize the 2/1 AP relative to the 1/1 AP, enabling the synthesis of the 2/1 AP from the vast knowledge on the 1/1 APs.⁶⁸⁾ Figure 8 shows X-ray diffraction patterns of the Au–Cu–Al–In–Gd system, as a typical example of the effect of isovalent substitution, which shows that the simultaneous substation of Cu and In for Au and Al, respectively, stabilizes the 2/1 AP relative to the 1/1 AP. Such stabilization of the 2/1 AP with multielement substitution may be attributed to the existence of larger number of crystallographic nonequivalent sites that can accommodate different atomic species for the higher-order approximant. Thus, the isovalent substitution is likely to be one effective way to synthesize even QCs and the search for magnetic QCs are now in progress.

Fig. 8

CuKα powder XRD patterns for Au₅₄Cu₉Al_9.5In_13.5Gd₁₄, Au₆₄Al₉In₁₃Gd₁₄, Au₅₉Cu₅Al₂₂Gd₁₄ and Au₆₄Al₂₂Gd₁₄, together with the results of Le Bail fitting. I_obs (red crosses) and I_cal (blue solid line) represent measured and calculated intensities, respectively. The purple and green vertical bars represent the Bragg peak positions for the space group $Im\bar{3}$ and $Pa\bar{3}$, respectively. The 2/1 approximants are obtained by simultaneous substitution of Cu and In for Au and Al, respectively, to the Au₆₄Al₂₂Gd₁₄ 1/1 approximant.⁶⁸⁾

5. Conclusion

The magnetism of R-containing Tsai-type 1/1 approximants were reviewed and discussed in terms of the average electron-per-atom (e/a) ratio. The net magnetic interaction characterized by the paramagnetic Curie-Weiss temperature Θ_p is found to be not much sensitive to the atomic species of the constituent elements but found to be well controllable by the e/a ratio. Also, the magnetic ground state can be well grouped into the ferromagnetic, antiferromagnetic and spin-glass regions according to the e/a ratio. In order to extend the magnetism research toward higher-order approximants including quasicrystals, isovalent substation has turned out to be one effective way for the synthesis of 2/1 approximants, which may also be applied to the synthesis of the first quasicrystals having long-range magnetic order.

Acknowledgement

This work was supported by JSPS KAKENHI Grant Numbers JP19H05817, JP19H05818.

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