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

Abstract

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.

Introduction

High demand for commercially viable high-energy permanent magnets stimulates active search for new prospective materials¹. New here also stands for “well-forgotten” old materials, which can now be upgraded (by elements substitution, tuning of composition, interstitial modifications, etc.) using deeper understanding of fundamental magnetic properties, stemming from modern ideas and techniques. It is the reason why the attention was recently returned to the compounds with the tetragonal ThMn₁₂-type of crystal structure for production of rare-earth-lean or rare-earth-free magnets^2–7. The iron-rich intermetallics RFe₁₁Ti, which demonstrate strong uniaxial magnetic anisotropy of the iron sublattice (with the first magnetocrystalline anisotropy constant (MCA) K₁ = 0.89 MJ/m³ at room temperature for YFe₁₁Ti)^8,9, are a good example. Interstitial modification of RFe₁₁Ti (with magnetic or non-magnetic rare-earth R ions) affect dramatically and usually positively main magnetic characteristics of the compounds such as Curie temperature (T_c), saturation magnetization (M_S), and the first MCA (K₁)^10–12.

The crystal lattice of RFe₁₁Ti contains a single 2a site (point symmetry 4/mmm) for the rare-earth atom and three non-equivalent Fe sites (8i, 8j and 8f). Notice that the 8i site is populated by a mixture of Fe and Ti⁹. In the case of interstitial modification by hydrogenation or nitrogenation, the H or N atoms occupy a single 2b site (see Fig. 1)^9,10.

Figure 1.

Schematic diagram representing position of hydrogen or nitrogen (arrows indicate the direction of magnetic moments for rare-earth atoms) in RFe₁₁Ti-H(N) compounds: left - for the hydride and nitride, right - for the parent alloy.

Rare earths and iron form two magnetic sublattices, which are either ferromagnetically ordered (co-aligned) in case of light rare-earths or ferrimagnetically aligned (anti-aligned) for heavy rare earths. The rather simple crystal structure of R(Fe,T)₁₂ or R(Fe,T)₁₂-X streamlines theoretical calculations of the CEF (CEF is the 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 RFe₁₁Ti compounds^13–16, as well as for the hydrided^16,17 and nitrided¹⁸ 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 ferrimagnets²⁰. 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 Tm³⁺ 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 TmFe₁₁Ti and TmFe₁₁TiH 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 TmFe₁₁Ti 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 H₂-gas hydrogenation procedure, single-crystallinity was preserved upon hydrogenation. Several samples of the TmFe₁₁Ti-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 TmFe₁₁TiH_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 TmFe₁₁TiN_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 N₂ was 1 ± δ atoms per RFe₁₁Ti 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 ms¹⁹. Magnetization study in steady magnetic fields was done using a standard PPMS 14 T magnetometer (Quantum Design, USA).

Results and Discussion

RFe₁₁Ti absorb a very small amount of hydrogen (a maximum of 1–1.1 at./f.u.) in contrast to e.g. R₂Fe₁₄B and R₂Fe₁₇ compounds, where the amount of absorbed hydrogen can reach 5 at./f.u.¹¹. XRD study showed that TmFe₁₁TiH_x (x = 0, 0.9, 1 and 1.1) and TmFe₁₁TiN_x (x = 1) retained the tetragonal ThMn₁₂ type (space group I4/mmm and Z = 2) of crystal structure after hydrogenation and nitrogenation. Lattice parameters are shown in Table 1.

Table 1.

Unit cell parameters of TmFe₁₁Ti-(H,N).

Compound	a, nm	c, nm	V, nm3	ΔV/V,%
TmFe11Ti	8.4655	4.7794	342.5	0
TmFe11TiH0.9	8.5013	4.7790	345.4	0.8
TmFe11TiH1	8.5053	4.7789	345.7	0.9
TmFe11TiH1.1	8.5093	4.7789	346.0	1
TmFe11TiN1	8.5542	4.8157	352.4	2.8

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%.

Figure 2 shows the magnetization curves at 4.2 K for the single crystalline TmFe₁₁Ti and TmFe₁₁TiH samples and for the aligned powder sample TmFe₁₁TiN in magnetic fields applied along the [001] axis. It is seen, that the ferromagnetic state is reached in the magnetic field of 52 T for TmFe₁₁TiH. Magnetic fields exceeding 60 T are needed to observe a ferromagnetic state both in the parent and in the nitrided compounds. It was shown earlier that Tm₂Fe₁₇²⁴ and Tm₂Fe₁₄B²¹ reach the ferromagnetic state in fields exceeding 80 T. It is clear that nitrogenation enhances the intersublattice exchange interaction in TmFe₁₁Ti.

Figure 2.

Magnetization curves of TmFe₁₁Ti, TmFe₁₁TiH and TmFe₁₁TiN measured at 4.2 K in pulsed fields applied along the [001] axis (dots represent measurements in steady magnetic fields).

Figures 3 and 4 demonstrate the magnetization curves of TmFe₁₁Ti and TmFe₁₁TiH at 4.2 K taken along crystallographic directions [001], [100] and [110].

Figure 3.

Theoretical (dashed lines) and experimental (solid lines) magnetization curves obtained for the TmFe₁₁Ti single crystal in magnetic fields applied along the [001] and [100] axis at 4.2 K.

Figure 4.

Theoretical (dashed lines) and experimental (solid lines) magnetization curves obtained for the TmFe₁₁TiH single crystal in magnetic fields applied along the [001] and [100] axis at 4.2 K.

While the four-fold symmetry axis c ([001]) is the easy magnetization direction, the two basal-plane curves [100] and [110] (not shown) measured along the hard magnetization directions look practically identical due to a weak anisotropy within the basal plane. In the parent TmFe₁₁Ti compound, the basal-plane magnetization curve crosses the c-axis magnetization at about 9 T, running above it close together in the interval from 9 T to 60 T. The TmFe₁₁TiH hydride shows only one step on the magnetization curve at 45 T. As a result, TmFe₁₁TiH proceeds from the ferrimagnetic state (where the total magnetization Ms = M_Fe − M_R = 13.8 µ_B/f.u) directly to the forced ferromagnetic state, with M_S = M_Fe + M_R = 27.8 µ_B/f.u.

The magnetic behavior of the TmFe₁₁Ti-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^15,25–27. 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

$$\begin{array}{lll}\hfill F& \hfill =\hfill & -N{k}_{B}T\log Z-M_{Fe}\left(H_x\sin \theta \cos \phi +H_y\sin \theta \sin \phi +H_z\cos \theta \right)\hfill \\ \hfill & \hfill & +K_1{\sin }^2\theta +K_2{\sin }^4\theta .\hfill \end{array}$$ 1

where $\mathrm{Z}={\sum }_{\mathrm{n}}\exp \left(\frac{-{\mathrm{E}}_{\mathrm{n}}}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right)$, E_n is the energy of the n-th level of the R ion. The second term of Eq. (1) represents the energy of the iron sublattice, the angles θ and φ are polar coordinates of the iron sublattice magnetization with respect to the main crystallographic directions ([001] and [100]), H = (H_x, H_y, H_z) is the external magnetic field, K₁ and K₂ are the anisotropy constants of the second and fourth order. Numerical values of the polar coordinates Θ and φ are obtained by minimizing the total free energy at given conditions (temperature, direction and magnitude of external magnetic field).

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 Tm³⁺ multiplet) and is given by:

$$H=H_{CF}+g_J{\mu }_B\mathit{J}\left({\mathit{H}}_{ex}+\mathit{H}\right)$$ 2

where g_J is the Landé factor, H_ex is the exchange field.

The crystal-field Hamiltonian is:

$$H_{CF}=B_0^2{C}_0^2+B_0^4{C}_0^4+B_0^6{C}_0^6+B_4^4(C_{-4}^4+C_4^4)+B_4^6(C_{-4}^6+C_4^6)$$ 3

with the crystal-field parameters ${\mathrm{B}}_{\mathrm{q}}^{\mathrm{k}}$ and single-electron irreducible tensor operators ${\mathrm{C}}_{\mathrm{q}}^{\mathrm{k}}={\sum }_{\mathrm{i}}{\mathrm{C}}_{\mathrm{q}}^{\mathrm{k}}(\mathrm{i})$.

The magnetization behavior of the system is obtained by using the following expression

$$M_{\alpha }=-\frac{\partial F}{\partial H_{\alpha }},\alpha =x,y,z$$ 4

The total magnetization of the parent TmFe₁₁Ti was estimated as Ms = M_Fe − M_R = 13.3 µ_B/f.u and this value is in good agreement with experiment. Estimated value of magnetization in the ferromagnetic state is M_S = M_Fe + M_R = 27.3 µ_B/f.u. The exchange fields H_ex for TmFe₁₁Ti and TmFe₁₁TiH₁ obtained by fitting our experimental magnetization data are 50.8 and 47.5 T, respectively. The CEF parameters for both compounds are presented in Table 2.

Table 2.

CEF (in cm⁻¹) and exchange (in T) parameters for TmFe₁₁Ti and TmFe₁₁TiH obtained by fitting the experimental magnetization data in the present work.

Compound	${\mathit{B}}_{\mathbf{0}}^{\mathbf{2}}$	${\mathit{B}}_{\mathbf{0}}^{\mathbf{4}}$	${\mathit{B}}_{\mathbf{0}}^{\mathbf{6}}$	${\mathit{B}}_{\mathbf{4}}^{\mathbf{4}}$	${\mathit{B}}_{\mathbf{4}}^{\mathbf{6}}$	H ex
TmFe11Ti	−17.1	−2.21	43.92	−23.98	0	50.8
TmFe11TiH	−50.2	−40.02	43.92	−23.98	0	47.5

CEF and exchange parameters of the parent TmFe₁₁Ti 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 $B_0^2,B_0^4,B_0^6$ and H_ex. The calculated CEF and exchange parameters for different hydrogenated samples are very close to each other because of the not large span of hydrogen content in the studied samples (x ranges between 0.9 and 1.1). However, there is a significant difference in the values of parameters between parent and hydrogenated compounds, especially, of $B_0^2,B_0^4$. The parameters for the TmFe₁₁TiH hydride are obtained for the first time.

The changes in the $B_0^2$ and exchange parameter after hydrogenation of TmFe₁₁Ti 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 HoFe₁₁TiH¹⁷, 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 T²². 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 TmFe₁₁Ti-X hydrides and a nitride in high magnetic fields. The main feature of our work is that in the case of a hydrided TmFe₁₁Ti 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 TmFe₁₁Ti 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 RFe₁₁Ti 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 atoms⁹. 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 problem¹⁶ 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 materials⁵. 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.

Author Contributions

I.S.T. designed the study and supervised the project; I.A.P. obtained initial compound TmFe₁₁Ti; N.V.K. and A.K.Z. performed theoretical investigations; E.A.T. collected and analyzed the magnetization data; M.P. and L.H. obtained hydrides with H content x ≥ 1 at.H/f.u.; H.D. obtained hydrides with H content x < 1 at.H/f.u.; Y.S. and M.D. performed high magnetic field measurements. All authors contributed to the interpretation of the data and commented on the manuscript.

Competing Interests

The authors declare no competing interests.