ThMn12-type phases for magnets with low rare-earth content: Crystal-field analysis of the full magnetization process

Rare-earth (R)-iron alloys are a backbone of permanent magnets. Recent increase in price of rare earths has pushed the industry to seek ways to reduce the R-content in the hard magnetic materials. For this reason strong magnets with the ThMn12 type of structure came into focus. Functional properties of R(Fe,T)12 (T-element stabilizes the structure) compounds or their interstitially modified derivatives, R(Fe,T)12-X (X is an atom of hydrogen or nitrogen) are determined by the crystal-electric-field (CEF) and exchange interaction (EI) parameters. We have calculated the parameters using high-field magnetization data. We choose the ferrimagnetic Tm-containing compounds, which are most sensitive to magnetic field and demonstrate that TmFe11Ti-H reaches the ferromagnetic state in the magnetic field of 52 T. Knowledge of exact CEF and EI parameters and their variation in the compounds modified by the interstitial atoms is a cornerstone of the quest for hard magnetic materials with low rare-earth content.

crystalline-electric-field acting on the rare-earth ion) and exchange interaction parameters which can be acquired by analyzing experimental magnetization curves obtained using standard techniques in steady magnetic fields. The examples of the crystal-field analysis can be found in literature for the parent RFe 11 Ti compounds [13][14][15][16] , as well as for the hydrided 16,17 and nitrided 18 series. Unfortunately, literature data show rather scattered values even within the same series, thus calling for a reliable solution. In order to obtain true CEF and exchange parameters, high magnetic fields should be employed. New experimental techniques allow determination of magnetization in high pulsed magnetic fields up to and above 60 T (with the maximum of 100 T) 19 . Such magnetic fields enable execution of a full magnetization process (i.e. magnetization all the way up to the forced-ferromagnetic state) in ferrimagnets 20 . A particular advantage is to perform such experiments using thulium compounds, since Tm has the Landé factor closest to unity, which allows reaching the ferromagnetic state in relatively weak magnetic fields 21,22 . The second-order CEF parameter at the 2a rare-earth site is negative and Tm 3+ having a positive second-order Stevens' coefficient α J strengthens the uniaxial anisotropy of the Fe sublattice. Forced-ferromagnetic state can be reached faster (in lower field) if the sample is magnetized along the easy magnetization direction (EMD). Moreover, hydrogen atoms introduced into the crystal lattice of the sample may in general reduce the R-Fe intersublattice exchange interaction thus lowering the field, at which a compound becomes ferromagnetic 22,23 . The purpose of this paper is to calculate to high accuracy the fundamental CEF and exchange interaction parameters in the TmFe 11 Ti and TmFe 11 TiH single crystals from the high-field magnetization measurements and to demonstrate how to control these parameters by modification of the structure with light interstitial elements.

Experimental details
Polycrystalline TmFe 11 Ti samples were prepared by arc melting of 99.95% pure elements under an argon atmosphere. The ingots were re-melted several times and then heated and cooled slowly in a resistance furnace. Needle-like single crystals (0.7 mm long) were extracted from the ingot and checked by the X-ray Laue technique. Using gentle H 2 -gas hydrogenation procedure, single-crystallinity was preserved upon hydrogenation. Several samples of the TmFe 11 Ti-H system were obtained with different hydrogen concentration close to the upper limit of hydrogen absorption for these materials. The amount of absorbed hydrogen in TmFe 11 TiH x (x ≈ 0.9, 1 and 1.1) was determined with an accuracy of ±0.02 from the hydrogen pressure change in the calibrated reactor chamber after finishing the reaction. The nitride TmFe 11 TiN x-δ was formed by blowing high-purity nitrogen gas at atmospheric pressure through fine powder samples (grains size less than 10 µm) at 500 °C for 4 h. The nitrogen content was estimated by determining the weight difference of the sample before and after the nitrogenation procedure. The amount of absorbed N 2 was 1 ± δ atoms per RFe 11 Ti formula unit (δ ≈ 0.05). The nitrided powder was fixed in epoxy resin in a magnetic field of 10 kOe to form aligned samples of cylindrical shape. Powder X-ray diffraction (XRD) analysis was used to determine the structure both of the parent compound and its hydrides and the nitride.
Magnetization measurements were performed using a pulsed-field induction magnetometer at the Dresden High Magnetic Field Laboratory. The maximum field was equal to 60 T and the total pulse duration was 25 ms 19 . Magnetization study in steady magnetic fields was done using a standard PPMS 14 T magnetometer (Quantum Design, USA).

Results and Discussion
RFe 11 Ti absorb a very small amount of hydrogen (a maximum of 1-1.1 at./f.u.) in contrast to e.g. R 2 Fe 14 B and R 2 Fe 17 compounds, where the amount of absorbed hydrogen can reach 5 at./f.u. 11 . XRD study showed that TmFe 11 TiH x (x = 0, 0.9, 1 and 1.1) and TmFe 11 TiN x (x = 1) retained the tetragonal ThMn 12 type (space group I4/mmm and Z = 2) of crystal structure after hydrogenation and nitrogenation. Lattice parameters are shown in Table 1.  The magnetic behavior of the TmFe 11 Ti-H system was described through quantum theory analysis, using a two-sublattice (rare-earth and iron sublattices) approximation for the magnetic structure and taking into account the exchange and CEF interactions. The method of theoretical investigation is essentially the same as described in refs 15,[25][26][27] . It is a general method of calculating magnetic properties in the R-M-X systems, where R and M are the 4f and 3d-transition metals and X is a non-magnetic element such as boron. In brief, the total free energy in this model has the form  N). One can see that hydrogenation leads to an increase of the a parameter while c slightly decreases. Volume expansion of the hydrides does not exceed 1%. Nitrogenation increases the relative unit cell volume ΔV/V by 3%. Hamiltonian of the rare-earth ion is Hermitian matrix with the dimension (2J + 1) × (2J + 1) (J is the total angular momentum of the ground Tm 3+ multiplet) and is given by: where g J is the Landé factor, H ex is the exchange field. The crystal-field Hamiltonian is: with the crystal-field parameters B q k and single-electron irreducible tensor operators = ∑ C C (i) q k i q k . The magnetization behavior of the system is obtained by using the following expression   Table 2.
CEF and exchange parameters of the parent TmFe 11 Ti were already obtained in refs 13,28 in fields up to 30 T. Our investigations in the magnetic fields up to 60 T allowed us to obtain redefined values of these parameters, especially of B B B , , The changes in the B 0 2 and exchange parameter after hydrogenation of TmFe 11 Ti attract a special attention. B 0 2 increases almost by a factor of 3 leading to a significant strengthening of uniaxial anisotropy 29,30 . The exchange field H ex slightly decreases from 50.8 to 47.5 T. Note, a slight increase in the R-Fe exchange interaction was reported previously for HoFe 11 TiH 17 , for which the studies were performed in the magnetic field much lower than the field of the transition from ferri-to the ferromagnetic state. This H ex parameter is responsible for reaching of the forced-ferromagnetic state in fields lower than 60 T 22 . Using the obtained full set of parameters we were able to calculate theoretical magnetization curves up to 100 T: it is especially important for the parent compound where the transition to the ferromagnetic state occurs in the magnetic fields (near 70 T) slightly exceeding the applied 60 T limit.

Summary
We have studied the magnetic properties of TmFe 11 Ti-X hydrides and a nitride in high magnetic fields. The main feature of our work is that in the case of a hydrided TmFe 11 Ti we were able to conduct theoretical analysis using a full magnetization process obtained experimentally. We also established that introduction of 1 H at./f.u. into TmFe 11 Ti weakens the intersublattice exchange despite the increasing magnetic moment on Fe atoms or, in other words, despite the strengthening of the Fe sublattice. Based on the results obtained we can provide the following recommendations. The obtained second-order crystal-field parameter B 0 2 for RFe 11 Ti is negative and small, but its value can be easily controlled by changing environment of the rare-earth ion with the aid of interstitial (Fig. 1) and by substitution atoms 9 . Here, we solve a direct problem, namely, we determine parameters of the crystal and exchange field from experimental magnetization curves using single-crystalline samples. It will also be possible to solve inverse problem 16 since R-Fe crystal field parameters do not change significantly within one series of compounds with various Rs. It will enable design and simulation (see e.g. studies predicting new materials) 31,32 of compounds with desired magnetic properties when we substitute an expensive rare-earth by cheaper R ions (for example, cerium) and/or other elements (for example, zirconium). Indeed, zirconium is already widely used for R-lean magnetic materials 5 . Promising magnetic characteristics were demonstrated e.g. for R(Fe,T) 12 (where R = Nd) compounds interstitially modified by nitrogen 3,18 . This gives hope that strong permanent magnets with low rare-earth content may soon become a reality.  Table 2. CEF (in cm −1 ) and exchange (in T) parameters for TmFe 11 Ti and TmFe 11 TiH obtained by fitting the experimental magnetization data in the present work.