High-field magnetization studies and their analysis in RFe₁₁Ti and RFe₁₁TiH₁ rare-earth intermetallics (an example: HoFe₁₁TiH_x, x = 0 and 1)

Hard permanent magnets based on rare-earth intermetallic compounds have a lot of perspective practical applications.^1–4 R(Fe,M)₁₂ compounds with ThMn₁₂ structure (M is a structure stabilizing element) are being rediscovered as the materials for creating new magnets with replacement possibility of the high-performance magnets based on Nd-Fe-B -type in order to expand possibilities for modifying properties (including light interstitial dopings) and limit costs.^5–7 This ensures the persistent and fundamental interest to the R-3d transition-metal compounds.^8–11 

There are three areas of research for the RFe₁₁Ti-type compounds. First, compounds with light rare-earths, whose magnetic moment is co-directional with the magnetic moment of iron sublattice are being studied. Currently, such compounds are used as good permanent magnets.^2–4,11 Secondly, the study of compounds of the (R,R’)Fe₁₁Ti with two different rare-earths is involved. This area is practically unexplored at present, but the example of well-known (R,R’)₂Fe₁₄B compounds demonstrates the prospects of such research, for instance, to increase the magnetic anisotropy or coercivity.^2,12,13 The third case, which is the most promising from the point of view of fundamental research and the search for new materials, is the RFe₁₁Ti-type compounds with heavy rare-earth ions. These compounds are investigated in this work. In this case, antiparallel coupling between the iron and rare-earth magnetic moments is realized, so they are ferrimagnets in small magnetic fields. For this reason, the influence of an external magnetic field on single-crystal samples is a powerful tool to study the exchange interactions and anisotropy, and extract magnetic parameters by revealing pecularities in the magnetization process.^8,14–17 Thus, RFe₁₁Ti compounds can be studied most effectively by magnetization measurements at sufficiently high magnetic fields as the ferrimagnetic structure is affected by the applied field, providing direct information on the intersublattice R-Fe exchange interaction and the magnetic anisotropy of the rare earths. Interstitial modification of the crystal lattice of these compounds is known to be a useful tool to improve both the Curie temperature and the saturation magnetization. An obvious exclamation for this is that increase of interatomic distances reduces the unfavorable direct exchange interaction within the 3d sublattice. Hence, in present work we consider the hydrides of the parent compounds, too. The second important method to identify properties suitable for the desired applications and to design materials based on them are the computational simulations.^18,19 Generally, the single-ion model for the crystal-electric-field interaction and a mean-field approximation for exchange interaction are often used to explain the experimental results. Examples of previous studies of RFe₁₁Ti can be found in the literature.^4,8,19,20 Unfortunately, magnetic parameters reported in the literature are often unable to describe correctly the complete transition from the ferri- to the forced ferromagnetic phase, that occurs in high magnetic fields (usually 50-80 T for 1-12 systems). This is due to the fact that the crystal-field parameters have often been calculated using experimental data obtained at lower magnetic fields than those required for the transition to the ferromagnetic phase. Therefore, in order to obtain correct values of intrinsic parameters of the compounds, sufficiently high magnetic fields should be used.²¹ 

Theoretical calculations can explain the behaviour of the magnetocrystalline anisotropy, which is one of the most important intrinsic magnetic features. In the HoFe₁₁Ti compound, the easy magnetization axis remains along the c-axis in the whole range of the magnetic order.²² At the same time, the strong rare-earth anisotropy in the ab-plane causes a fundamentally different behavior of the magnetization curves along the [100] and [110] main crystallographic axes. This allows to obtain more useful information about the internal parameters of the material from the high-field experimental data.

In the present work, we perform complete and accurate investigations of the high-field magnetization behavior up to 80 T and evaluate a complete set of the crystal-field and exchange parameters for HoFe₁₁Ti and HoFe₁₁TiH₁, also discussing the obtained results in comparison with the erbium and thulium compounds. We investigate HoFe₁₁Ti and its hydride, for which the experimental high-field data have been obtained in fields up to 60 T²⁰ and calculate the high-field magnetization behavior.

The HoFe₁₁Ti samples were obtained by induction melting of the constituent elements with a purity of at least 99.95 wt.% under an argon atmosphere as described in Ref. 20. The single crystals weighing a few milligrams were then taken from the crushed ingots. The single-crystalline state of the samples was checked using a conventional back Laue reflection method. The samples were then hydrogenated in a stainless steel chamber. A special temperature program has been applied in order to preserve the single-crystalline state of RFe₁₁Ti after hydrogenation. A short thermal activation process at 420°C in high vacuum was carried out to initiate the hydrogen absorption process and was completed by cooling down to 20°C for one hour. After that, the reaction chamber was filled with high-purity hydrogen gas under 1-3 atm pressure and the system was heated to 350°C. The samples were annealed in the hydrogen atmosphere for 12 hours and slowly cooled down to room temperature. The hydrided RFe₁₁TiH₁ single crystals remained of good quality, as confirmed by X-ray backscattering Laue patterns.

The high-field magnetization measurements at 4.2 K were performed in pulsed magnetic fields up to 60 T using the equipment in the High-Field Laboratory at Dresden-Rossendorf. The pulsed-field data were calibrated using magnetization curves measured on the same samples in static magnetic fields up to 14 T in a commercial PPMS14 magnetometer (Quantum Design, USA).²³ All magnetization curves presented below were corrected for demagnetizing field.

High-field data on HoFe₁₁Ti was published earlier by members of our group,²⁰ so we will use this data further in the text. Figures 1, 2 and 3 show the experimental magnetization curves of an HoFe₁₁Ti and its hydride HoFe₁₁TiH₁ single crystals at 4.2 K in a magnetic field applied along the [001], [100] and [110] crystallographic directions, respectively, and the theoretical curves shown by the dashed line. We discuss the numerical calculations in the next theoretical part of the article. Along the [001] crystallographic direction, the magnetization type is easy axis for HoFe₁₁Ti compound, moreover, in a magnetic field more than 40 T, a first-order transition to the angular phase is observed. In the high-field region, magnetization jumps are observed at 45 and 53 T along the [001] and [100], respectively, for HoFe₁₁Ti and 28 and 55 T for HoFe₁₁TiH₁. The magnetization measured for HoFe₁₁Ti along the [110] direction shows a small kink at about 50 T²⁰ and a smooth curve for its hydride. The difference in magnetization curves along two directions in the basal plane is quite significant, and confirms surprisingly strong in-plane anisotropy for both the hydride and the parent compound. One can see the beginning of very broad transition from ferri- to ferromagnetic phase along all main crystallographic directions. This transition for the [001] and [100] directions takes place in several sharp steps between 25 and 60 T and 40 and 60 T, respectively. The experimental curves along the [110] direction are more smooth for both pure and hydrogenated compound.

RFe₁₁Ti can be considered as a magnet with two-sublattices in which the R and Fe magnetic moments are ordered either ferromagnetically (co-aligned) in the case of light rare-earths or ferrimagnetically (anti-aligned) for heavy rare-earths. To analyze the high-field experimental magnetization data, we use a method that we have successfully implemented to calculate the magnetic properties of the ferrimagnetic intermetallic systems with rare-earths (with the localized 4f electrons) and iron (with the itinerant magnetic electrons) sublattices.^7,21

The total Hamiltonian of the holmium rare-earth ion is of the form:^21,23

where g_J is the Landé factor, H_ex is the R-Fe exchange field, H is external magnetic field, $HCF$ is CEF (crystal-electric-field) Hamiltonian. The Ho³⁺ ions in RFe₁₁Ti possess the tetragonal site symmetry and hence the CEF hamiltonian is given by

here $Bqk$ are CEF parameters and $Cqk=∑iCqki$⁠, and $Cqk(i)$ are single-electron irreducible tensor operators. It is well known, that it is sufficient for the description of the macroscopic magnetic properties with high accuracy to consider only the ground state of the rare-earth ion’s multiplet.^7,21,23

The total free energy of the R-Fe system is:^7,21,23

where $ZHo=∑n−EnkBT$⁠, E_n are energy levels of the holmium ion derived for the CEF Hamiltonian by diagonalization (see Eq. (2)), K₁ and K₂ are second and fourth order anisotropy constants of the iron sublattice, respectively,^8,N is the number of rare-earth ions, M_Fe(T) is the spontaneous magnetization. The compounds with the nonmagnetic rare-earths Y and Lu can be used to evaluate the iron-sublattice contribution.^24,25 The values of $MFe0$ (at zero temperature) reported in the literature are slightly different, varying from 19.4μ_B/f · u.²⁵ to 21μ_B/f · u.²⁰ In this work, we use the second value as a more accurate and new value in our estimation.

Numerical values of polar angles θ and φ of iron sublattice magnetization with respect to the main crystallographic directions are calculated by minimizing F(θ, φ) at given conditions: temperature and external magnetic field.

The magnetization has the form:

Theoretical magnetization M(H) curves for HoFe₁₁Ti and its hydride in fields have been calculated. A complete set of the CEF and exchange parameters obtained by comparing theoretical and experimental high-field magnetization curves is presented in Table I. The literature parameters given for comparison are taken from Refs. 20, 22, and 25. It can be seen, that measurement up to quite moderate magnetic fields are sufficient to obtain the sign of some crystal-field parameters. However, for an accurate theoretical description of high-field magnetization curves, parameters obtained from measurements in higher magnetic fields (for example, up to 60 T) are required. Using previous research in weaker magnetic fields, we have made a revision of the crystal-fields and exchange parameters. The agreement with the experimental magnetization data is satisfactory (see Figs. 1–3), confirming the relevance of the estimation of our parameters.

This kind of studies is extremely important, as it provides us with information about both field and temperature behavior of materials, which are necessary for various practical applications. Understanding the behavior and intrinsic nature of rare-earth-based materials is key to modifying them for ever-growing technical and practical requirements.

In this work, too, we again observe the common behaviour of all ThMn₁₂-compounds concerning the R-Fe interaction feature: the exchange coupling decreases after hydrogenation.^6,7,21,23,26 For HoFe₁₁Ti decreasing of the H_ex value is significant and amounts to 40 T, which exceeds the values for compounds with erbium and thulium ions (see Table II). Such a significant reduction is obviously very promising for various practical applications.

In comparison to others, values of the first three crystal-field parameters are larger for Ho³⁺ ions (both in pure and hydride compounds), which is due to significant rare-earth Ising Ho³⁺ ion anisotropy. Furthermore, we can see that hydrogenation does not change the sign of the first three crystal-field parameters, as the type of the anisotropy does not change.

In this work, we use the combined experimental and theoretical efforts to understand intrinsic magnetic properties of the HoFe₁₁Ti and its hydride HoFe₁₁TiH₁ by calculating the crystal-field and exchange parameters. High-field magnetization has again been demonstrated to be a useful tool for studying the intersublattice exchange in rare-earth ferrimagnets. Using the evaluated intrinsic parameter values we construct theoretical magnetization curves for HoFe₁₁Ti and its hydride HoFe₁₁TiH₁ and describe a set of field-induced transitions. In this way, we were able to observe the full magnetization process along the main crystallographic directions. The results were compared with compounds containing erbium and thuliums ions.

Physical properties measurements were performed in the Materials Growth and Measurement Laboratory (http://mgml.eu/) supported within the program of Czech Research Infrastructures (project no. LM2018096). We acknowledge the support of HLD at HZDR (member of the European Magnetic Field Laboratory).

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article.