Speromagnetism and asperomagnetism as the ground states of the Tb-Dy-Ho-Er-Tm “ideal” high-entropy alloy
Introduction
An ideal high-entropy alloy (HEA) is by definition a solid solution of five or more chemical elements, distributed randomly on an undistorted crystal lattice. Such topologically ordered, but chemically (substitutionally) disordered system can be conveniently termed as a “metallic glass on an ordered lattice”, possessing simultaneously the properties of an ordered crystal and an amorphous glass. A good approximation of an ideal HEA is a mixture of rare earth (RE) elements with zero pair mixing enthalpies between any pair of the elements i and j, 0, that leads to completely random mixing of the elements [[1], [2], [3], [4]]. Extended solid solubility of the elements is augmented by choosing elements with the same crystal structure [5], whereas minimal lattice distortions are achieved by taking elements with very similar atomic radii , so that the atomic-size-difference (geometric) parameter is minimized. Here is the number of components, is the molar concentration of the element i and is the composition-averaged radius. A prototype of an ideal HEA is an equiatomic solid solution Tb-Dy-Ho-Er-Tm, composed of the elements from the heavy half of the lanthanide series. Pure metals all possess a hexagonal close-packed (hcp) structure at room temperature (RT), their pair mixing enthalpies are zero [6], and, being neighbors in the periodic system, the geometric parameter assumes the smallest value ( 0.5%) of all HEAs reported so far in the literature, so that the lattice distortions are minute. In our study, we have synthesized a Tb20.3Dy20.7Ho20.3Er19.7Tm19.0 polycrystalline material of hexagonal symmetry (hcp structure, space group P63/mmc, lattice parameters a = 3.582(2) Å and c = 5.632(3) Å). Details of the material synthesis, the XRD (Fig. A1) and the SEM (Fig. A2) characterization are given in Appendix A, whereas the properties of pure elements and metals are collected in Table A1 of Appendix A. The composition-averaged theoretical lattice parameters of this alloy = 3.575 Å and = 5.622 Å match almost perfectly the experimental ones, supporting random mixing of the elements, whereas minute value of the geometric parameter 0.48% for this composition indicates an almost undistorted hcp lattice. The Tb-Dy-Ho-Er-Tm HEA (abbreviated in the following as TDHET) is thus an excellent physical realization of a metallic glass on a topologically ordered lattice.
Triply ionized elements Tb, Dy, Ho, Er and Tm all possess large, localized 4f magnetic moments of paramagnetic values 9.72, 10.65, 10.61, 9.58, and 7.56 (in units of Bohr magnetons ), so that TDHET is characterized by a random distribution of large atomic moments of different sizes on a hcp lattice, with all sites being magnetic. In a metallic environment, the localized 4f moments couple indirectly via the electrons in the 5d/6s conduction band, leading to the long-range oscillatory RKKY coupling between the moments. The effective coupling between two localized moments separated by is , where is the sf exchange integral, is the number of conduction electrons per atom, is the RKKY function with (where is the Fermi wavevector) and is the Fermi energy. Since is typically of the order 0.1 nm−1, the sign of fluctuates on a scale of nm, with many shells of interacting neighbors having either positive or negative coupling. The RKKY coupling constant of two RE ions of angular momentum and Landé factor is , where is the de Gennes factor. For unlike ions () and (), the de Gennes factor is modified to [7]. It is known that the magnetic ordering temperatures of any series of RE metals or compounds with the same electronic band structure and similar lattice spacing scale with (or ), reflecting scaling of the coupling constant by the same factor. In the TDHET lattice, all five types of magnetic ions experience the same band structure and lattice spacing, so it is straightforward to assume that the RKKY coupling between the ions i and j can be written as . Taking the values of the five ions Tb, Dy, Ho, Er and Tm collected in Table A1, we show in Fig. 1a the de Gennes factor normalized to for all fifteen possible atomic pairs. An almost continuous distribution is observed from (the largest) to (the smallest, amounting to 11% of ). The first coordination shell of a given atom in the hcp structure of P63/mmc symmetry is shown in the inset of Fig. 1a. Each atom lies in a hexagonal plane normal to the z-axis and has 12–fold coordination, with six neighbors in the same plane, and three neighbors each in the upper and lower planes. The central atom is RKKY-coupled to all neighbors, and there is an enormous number of possibilities on how to distribute five different elements randomly on the twelve sites of the first coordination shell. This indicates that in the TDHET crystal, there exists an enormous number of different local chemical environments and the exchange interaction is distributed with a broad distribution function (dropping the subscript RKKY), which can be considered symmetric (e.g. Gaussian-like) around the mean value , with a variance .
The triply ionized ions Tb, Dy, Ho, Er and Tm also possess markedly different single-ion magnetocrystalline anisotropy, originating from the electrostatic interaction of the 4f charge cloud with the crystal electric fields (CF). The CF interaction stabilizes a particular electronic orbital, whereas the spin-orbit interaction then leads to magnetic moment alignment in a specific crystallographic direction. The five types of ions differ in the shape of the 4f electron cloud, which is oblate (flattened) for Tb, Dy and Ho, and prolate (elongated) for Er and Tm. At RT, the dominant term in the single-ion anisotropy energy is the interaction of the 4f electric quadrupole moment with the electric field gradient created by the charge distribution of the crystal at the RE site. Since for oblate ions, the moments of Tb, Dy and Ho align in the hexagonal plane, whereas the moments of the prolate ions Er and Tm () align along the perpendicular hexagonal direction. At low temperatures, the hexadecapole and the 64-pole moment of the 4f charge density expanded in spherical harmonics also become important in the anisotropy energy, which pull the Tb, Dy, and Ho moments out of the hexagonal plane and the Er and Tm moments away from the hexagonal axis in a temperature-dependent manner. In the TDHET crystal, there exist local easy directions (energetically favorable directions) of the moments on the scale of an atom or a nanoscale volume, which wander randomly over a few nearest-neighbor distances, so that TDHET can be considered as a random local anisotropy system with a continuous distribution of local anisotropies , where the leading term in the anisotropy energy of the site i is . In 4f alloys, the single-ion anisotropy energy is usually sufficient to pin the local magnetization direction.
The magnetic dipolar interaction between the 4f localized moments in the chemically disordered TDHET lattice also varies from site to site, characterized by a distribution of the dipolar magnetic fields in both magnitude and direction.
The TDHET HEA thus represents a magnetically concentrated system with all sites magnetic, containing randomness (five different types of spins are placed randomly on the hcp lattice) and frustration (no spin configuration can satisfy all the bonds and minimize the energy at the same time). Unlike the site-ordered crystalline magnetic systems, which are typically characterized by a few discrete values of the atomic moments , the exchange interaction , the anisotropy and the dipolar interactions , the TDHET HEA is characterized by probability distributions , , , and , shown schematically in Fig. 1b. The cooperative magnetism of such a system, which generally belongs to the broad class of spin glasses [8], may be highly complex. Spin glasses contain a wide range of magnetically diluted and concentrated materials, crystalline or amorphous, site-disordered or site-ordered, but geometrically frustrated. The influence of the duality of an ordered crystal and an amorphous glass on the magnetic ground state of the TDHET “ideal” HEA is the basic question addressed in this paper.
Section snippets
Magnetic measurements
Magnetic measurements were performed on a needle-shaped sample with the long axis parallel to the magnetic field, in order to minimize the demagnetization effects. The direct current (dc) magnetization in a low magnetic field 0.8 mT, measured for the zero field cooled (zfc) and field cooled (fc) protocols, is shown in Fig. 2a ( is presented in Bohr magnetons per formula unit, i.e., per one Tb0.203Dy0.207Ho0.203Er0.197Tm0.190 “molecule”). Upon cooling from RT, a tiny maximum is first
Discussion
The nature of the collective magnetic state in the TDHET “ideal” HEA is determined by an interplay of the probability distributions of the atomic moments , the exchange interaction , the anisotropy , and the dipolar interaction . The dipolar interaction, which leads to the domain formation in magnetic structures with a net FM moment may be, to a first approximation, ignored in an exchange-dominated system of RE elements from the heavy half of the lanthanide series without
Conclusions
The Tb-Dy-Ho-Er-Tm HEA is a physical realization of an ideal HEA with completely random mixing of the elements on a practically undistorted hexagonal lattice. It represents a magnetically concentrated system with all lattice sites occupied by localized magnetic moments and containing randomness and frustration due to chemical disorder, sharing properties of an ordered crystal and an amorphous glass. We have studied the influence of this crystal-glass duality on the collective magnetic state and
Author statement
M. Krnel, S. Vrtnik, P. Koželj and Z. Jagličić performed magnetic and electrical measurements. A. Jelen performed electron microscopy measurements, whereas A. Meden conducted X-ray diffraction study. M. Feuerbacher has grown the alloy and co-leaded the project. J. Dolinšek has written the manuscript and co-leaded the project.
Declaration of competing interest
The Authors declare that they have no conflict of interests.
Acknowledgments
Slovenian authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0125). MF acknowledges financial support from the German Research foundation (DFG) under grant No. FE 571/4 within the priority program SPP2006 “Compositionally Complex Alloys – High Entropy Alloys (CCA-HEA)”.
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