**2. Materials and Methods**

#### *2.1. Sample Preparation*

The samples for this study (NdxDy1-x)2Fe14B with x = 0, 0.5 and 1 were prepared using an arc furnace. The procedure for the samples obtaining is described in more details in the following works [17,18]. The phase composition of the samples was determined using the standard X-ray powder diffraction (XRD) at room temperature. XRD studies indicated that the Nd2Fe14B and Dy2Fe14B alloys were single-phase, while, in the (Nd,Dy)2Fe14B samples, traces of the second phase were seen (~5%). The investigated alloys have a tetragonal structure of the Nd2Fe14B type (space group P42/mnm) at room temperature. The lattice parameters for Nd2Fe14B, (NdxDy1-x)2Fe14B and Dy2Fe14B alloys are a = 0.880 nm, c = 1.219 nm, a = 0.877 nm, c = 1.212 nm, a = 0.873 nm, and c = 1.190 nm, respectively. The decreasing lattice parameter for the Dy-substituted compound is due to the smaller atomic radius of Dy3+ as compared to Nd3+. This agrees well with literature data [16].

### *2.2. Method of Magnetization Measurements*

Magnetization measurements were performed at the Russian Federal Nuclear Center in Sarov in pulsed magnetic fields up to 170 T on powder samples. An ultrahigh magnetic field up to 600 T was created in a magnetocumulative generator MC-1 [19]. In a thin-walled wire solenoid, the discharge of a powerful capacitor bank with a stored energy of 2 MJ created a seed magnetic field of about 16 T. During the battery discharge, a converging shock wave was initiated in the annular charge of high explosives surrounding the solenoid. It came out on the surface of the solenoid at the seed field maximum (approximately 80 μs after the start of the discharge). When the shock wave passed through the solenoid, the solenoid wires were welded and formed a homogeneous conducting cylinder shell with the trapped magnetic flux. The ultrahigh magnetic field was generated by explosive magnetic flux compression for about 16 μs. The MC-1 generator has been widely used earlier and proved to be a reliable tool for scientific research [20]. The high uniformity of the magnetic field in large useful volumes (about 10 cm3) made it possible to install four researched samples in one experiment.

## *2.3. Registration of the Signal*

The registration of the time derivative of the magnetic field was carried out by a set of pick-up coils with different sensitivities (7 coils in total, some of them were duplicated to increase reliability). It allowed measurements of the magnetic field induction with an accuracy of 5% over the entire operating range of the MC-1 generator. The magnetization of the studied samples was measured using compensated pick-up coils [21,22]. A pair of two identical coils were of diameter *d* = 2.8 mm and had number of turns *N* = 20. A special winding of the sensor was performed, which provided a significant reduction of the total electrical voltage between the coils of the sensor [21]. The signal induced in the compensation coils consists of a "useful" part and background signal, which stemmed from the coil decompensation and was proportional to the time derivative of the magnetic field. The degree of decompensation for all sensors was less than 2%. To take into account a slight attenuation of the signals of the compensation sensors in cable lines, the whole system was pre-calibrated immediately before the experiment. The compensated pick-up coils were proven to be a reliable technique for magnetization measurements at pulsed magnetic fields [20–22]. The compensation sensors were placed in a glass helium cryostat, into which liquid helium was raised from a transport Dewar vessel before the experiment.

The absolute values of the magnetization were calibrated by measuring the magnetization curves up to 14 T in static fields using a PPMS 14T magnetometer (Quantum Design, San Diego, CA, USA).

#### **3. Results**

Figure 2 displays the magnetization curves of the Dy2Fe14B and Nd2Fe14B single crystals measured at 1.8 and 10 K, respectively. Measurements have been performed along the main crystallographic axes and compared with the data given in the works [18,23]. Nd2Fe14B samples display an easy-cone anisotropy, in contrast to the uniaxial Dy2Fe14B with an anisotropy field of 27 T. In the inset in Figure 2, we show the magnetization curve of aligned polycrystal sample Dy2Fe14B in magnetic fields up to 120 T obtained at 10 K [16]. The anomaly near 100 T (at 105 and 101 T for increasing and decreasing fields, respectively, which correspond to intermittent changes in the M(H) curve) arises from the first-order transition between a collinear ferrimagnet and a non-collinear spin-flop-like phase. In the Nd2Fe14B single crystal, a jump in the M(H) magnetization curve is observed along the [100] direction in magnetic field 17 T.

**Figure 2.** Magnetization curves of Dy2Fe14B and Nd2Fe14B single crystals applied along the main crystallographic directions at 1.8 and 10 K, respectively [18,23]. Inset: The data for an aligned polycrystal Dy2Fe14B at 10 K (for increasing fields [16]) are shown for comparison.

Figure 3 demonstrates the magnetization curves of the (Nd0.5Dy0.5)2Fe14B compound along the perpendicular [001] easy axis. Magnetization measurements were performed by us at the Dresden High Magnetic Field Laboratory in pulsed magnetic fields up to 58 T previously [18]. The measurements up to 170 T obtained for the first time at the Russian Federal Nuclear Center in Sarov. It should be mentioned that a compensated pick-up sensor, in fact, measured a time derivative of magnetization. This is why the flat segments of the magnetization curve in Figure 3 were beyond the sensitivity and shown by the dashed line. A partial substitution of Dy by Nd atom in the latter compound results in an increase of the anisotropy field where the easy- and hard-axis magnetization curves intersect. Such new results, containing features on the magnetization curve, can also be used for modern

numerical calculations, including for obtaining or refining the exchange and crystal-field parameters in the quantum model of the crystal electric field [16–18].

**Figure 3.** Magnetization curves of (Nd0.5Dy0.5)2Fe14B at 1.8 K in magnetic fields up to 58 T applied along the perpendicular easy axis c (red and black lines) [18] and at 4.2 K in fields up to 170 T (for increasing fields) applied along the c-axis (green and blue lines).

Figure 3 shows that the experimental data obtained in different laboratories are in good agreement with each other. Moreover, it can be stated that ultrahigh magnetic fields in the (Nd0.5Dy0.5)2Fe14B compound induce a magnetic phase transition similar to that observed earlier in the Dy2Fe14B compound. These transitions can be interpreted as intermediate processes from ferrimagnetism to forced ferromagnetism. A significant difference here is the fact that the magnetic phase transition in the substituted composition (Nd0.5Dy0.5)2Fe14B occurs smoothly, covering a quite large range of magnetic fields from 105 T to 160 T, exhibiting features characteristic of second-order phase transitions.

#### **4. Discussion**

In order to analyze obtained experimental data, it is important to understand which sections of the magnetization curves correspond to which phase: ferrimagnetic, non-collinear, or ferromagnetic (field-induced forced ferromagnetic state). This can be understood using a quite simple analytical approach in terms of the first and second critical fields. The first critical field *Hc*<sup>1</sup> corresponds to the transition from the ferrimagnetic to the non-collinear phase, and the second *Hc*<sup>2</sup> to the ferromagnetic one. In order to estimate them, we applied the analytical approach (previously described in detail in Refs. [24–27]) well proven for the (R,R')2Fe14B intermetallic compounds.

For the R2Fe14B compounds, both critical fields *Hc*<sup>1</sup> and *Hc*<sup>2</sup> are equal to [24,26,27]:

$$\begin{aligned} H\_{c1} &= \lambda \left( M\_{F\varepsilon} - 2M\_R \right) + \frac{H\_{\varepsilon} \, 2M\_R}{\lambda \left( M\_{F\varepsilon} - 2M\_R \right)}, \\ H\_{c2} &= \lambda \left( M\_{F\varepsilon} + 2M\_R \right) - \frac{H\_{\varepsilon} \, 2M\_R}{\lambda \left( M\_{F\varepsilon} + 2M\_R \right)} \end{aligned} \tag{1}$$

where *λ* is the R-Fe intersublattice exchange parameter, *Ha* = <sup>2</sup>*K*<sup>1</sup> *MFe* is the magnetic anisotropy field, and *K*<sup>1</sup> is magnetic anisotropy constant [5]. Here, the second term describing the anisotropy has been added in order to make a more accurate critical fields estimation [27]. Thus, the accuracy of analytical evaluations of the critical fields reaches several Tesla.

For the (Nd0.5Dy0.5)2Fe14B compound with two different rare-earth ions, the critical fields *Hc*<sup>1</sup> and *Hc*<sup>2</sup> have the form [27]

$$H\_{c1} = \lambda\_{Dy} \left(M\_{Fc} - 2M\_{Dy}\xi\_1\right) - \frac{2M\_{Dy}H\_{b\xi\_1^2}}{\left(M\_{Fc} - 2M\_{Dy}\xi\_1\right)'},$$

$$H\_{c2} = \lambda\_{Dy} \left(M\_{Fc} + 2M\_{Dy}\xi\_2\right) + \frac{2M\_{Dy}H\_{d\xi\_2^2}}{\left(M\_{Fc} + 2M\_{Dy}\xi\_2\right)'},\tag{2}$$

$$\xi\_i(H\_{ci}) = \frac{1}{1 + \lambda\_{Nd\xi\lambda M}(H\_{ci})};\ \chi\_{Nd}(H\_{ci}) = \frac{2M\_{Nd}}{\lambda\_{Nd}M\_{Fc} + H\_{ci}}; i = 1, 2$$

*λNd* and *χNd* are the exchange parameter and susceptibility of the Nd sublattice, respectively. In accordance with our previous estimates, *ξ<sup>i</sup>* ≈ 0.9 [24]. Formula (2) with anisotropic correction is universal and could be used for various rare-earth intermetallics. *Hc*<sup>1</sup> and *Hc*<sup>2</sup> values obtained for (Nd0.5Dy0.5)2Fe14B and Dy2Fe14B by analyzing high-field experimental data using Formulas (1) and (2) are given in Table 1.

**Table 1.** Magnetic critical fields' parameters *Hc*<sup>1</sup> and *Hc*<sup>2</sup> for (Nd0.5Dy0.5)2Fe14B and Dy2Fe14B.


*Hc*<sup>2</sup> value provides important information, which the magnitudes of external magnetic fields require to reach the ferromagnetic state, and allow us to plan new ultrahigh magnetization experiments for studying similar types of compounds. It can be seen that, in order to experimentally observe the transition to the forced-ferromagnetic phase, both compounds ((Nd0.5Dy0.5)2Fe14B and Dy2Fe14B) require magnetic fields greater than 250 T. This is confirmed, among other things, by the value of the magnetization in the (Nd0.5Dy0.5)2Fe14B compound at 170 T. It reaches magnitude 34 μB/f.u., while the magnitude of the magnetization in a forced-ferromagnetic state is 44.1 μB/f.u. (see Table 2), that is, in the magnetic field 170 T, saturation has not yet been obtained.

**Table 2.** Values of the rare-earth and iron magnetic moments for (Nd0.5Dy0.5)2Fe14B and Dy2Fe14B intermetallic compounds.


#### **5. Conclusions**

Magnetization measurements were performed for the intermetallic ferrimagnetic compound (Nd0.5Dy0.5)2Fe14B using ultrahigh magnetic fields up to 170 T. Such high magnetic fields make it possible to observe the experimentally field-induced phase transition from the initial ferrimagnetic state to the non-collinear one. The results obtained were compared with the literature high-field magnetization data for the basic Dy2Fe14B compound. We have determined two critical fields (*Hc*<sup>1</sup> and *Hc*2) of field-induced transitions for the compounds under study. We demonstrate that, in order to observe the transition to the forcedferromagnetic state, both compounds ((Nd0.5Dy0.5)2Fe14B and Dy2Fe14B) require magnetic fields greater than 250 T. It is shown that the replacement of one rare-earth atom by another is a powerful tool for controlling the properties of rare-earth R2Fe14B-type compounds, on the basis of which modern highly efficient permanent magnets are manufactured.

**Author Contributions:** Methodology, Y.B.K., D.A.M., V.V.P. and O.M.S.; investigation, N.V.K., A.K.Z., A.I.B., S.V.G., R.V.K., A.S.K., I.S.S., I.V.M. and A.V.F.; writing—original draft preparation, N.V.K.; writing—review and editing, I.S.T. and Y.B.K.; supervision, I.S.T. and Y.B.K.; project administration, A.K.Z., P.B.R. and V.D.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Ministry of Science and Higher Education of the Russian Federation (No. FSMG-2021-0005). Research in ultrahigh magnetic fields was carried out within the framework of the scientific program of the National Center for Physics and Mathematics (Project "Research in high and ultrahigh magnetic fields").

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
