Effect of Ce and Gd on the magnetic and mechanical properties of Nd2Fe14B permanent magnets

Mphamela Baloyi1, Ramogohlo Diale2 and Hasani Chauke1

1Materials Modelling Centre, University of Limpopo, Polokwane, South Africa

2MINTEK, Advanced Materials Division, Randburg, South Africa

E-mail: mphamela.baloyi@gmail.com

Abstract. Neodymium-based permanent magnets (Nd₂Fe₁₄B) can be used in various applications such as electric vehicles and wind power generators due to their high magnetic field strength and demagnetisation resistance. However, they suffer from low operating temperatures below 585 K and poor ductility. In this study, we investigate the effect of rare earth elements such as Gd and Ce on the Nd2Fe14B magnets. Their structural, electronic, mechanical and magnetic properties were calculated using the ab initio density functional theory approach. The partial substitution of Nd with Gd improves the ductility and mechanical stability of Nd2Fe14B permanent magnets. These findings provide insight into the future development of permanent magnetic compounds with good mechanical properties and high operation temperatures.

Introduction

A material that sustains its magnetic characteristics due to its inherent structure is called a permanent magnet (PM) [1]. Alnico, ferrite, ceramic, and rare-earth magnets are among the several types of magnets that are included in PM. Motors, generators, sensors, transducers, mechanical devices, lutches, and imaging systems are only a few of the many uses for these magnets [1]. Due to their severe brittleness and susceptibility to corrosion, rare earth magnets are typically coated or plated to stop corrosion [2]. Two well-known rare earth magnets are samarium cobalt (SmCo₅) and neodymium (Nd₂Fe₁₄B) due to their excellent magnetic properties. SmCo5 magnets possess a high Curie temperature of 993 K and are resistant to oxidation. However, they are less frequently used since they are more expensive and have a weaker magnetic field strength [3].

Among all commercial PM materials, sintered NdFeB magnets are the most commonly utilised due to their largest energy product[4]. Moreover, these magnets can retain a strong magnetic field once magnetised due to high remanence (Br). The demand for sintered Nd2Fe14B magnets has been rising annually due to the development of clean energy technology such as wind turbines and hybrid electric vehicles [5]. Applications such as energy motor vehicles and wind power generators require a material with a high Curie temperature above 473 K. However, previous studies indicated that Nd2Fe14B magnets possess poor mechanical properties (Poisson’s ratio) [6] and low Curie temperature [7]. In addition, these magnets display poor magnetocrystalline anisotropy at high temperatures, which suggests that a specific magnetic orientation of the magnet weakens as the temperature rises [8]. Neodymium is one of the key components in Nd2Fe14B magnets, contributing significantly to their magnetic strength. However, due to the limited availability of neodymium and the drawbacks of Nd2Fe14B permanent magnets, partial substitution of Nd with other available rare earth elements is considered.

Previous research on the impact of terbium on the magnetic characteristics of Nd2Fe14B magnets has shown that terbium raises Tc while decreasing magnetic moments to 19.20 µB [9]. By substituting Gd, Zheng et al.[10] examined the enhanced thermal stability of nanocrystalline melt-spun Nd-Pr-Fe-B alloys. In addition to being substantially less expensive than Dy and Tb and even less expensive than Nd and Pr, Gd raises the Curie temperature of the RE2Fe14B phase to 643 K [10]. It was also reported that the Ce2Fe14B magnet has a high magnetic strength of 22.70 µB and high saturation magnetisation of 1.16 T, which indicates that the material can be very strongly magnetised [3].

The thermodynamic, mechanical, and magnetic characteristics of ternary Nd2Fe14B, quaternary NdGdFe₁₄B, and NdCeFe14B permanent magnetic alloys are investigated using first principle density functional theory (DFT) calculations. It was found that the Gd replacement improved the ductility of Nd2Fe14B magnets. The crystal structures of Nd2Fe14B doped magnetic alloys with a space group of P4_2/mnm are shown in Figure 1. Ce and Gd are rare earth elements (RE) that substitute for Nd.

RE Fe Nd B
Figure 1: Crystal structure of Nd2Fe14B doped permanent magnets.

Computational Method

The Vienna ab initio simulation package (VASP) [11] incorporated the ab initio DFT [12] formalism, which was used to perform the structural, magnetic, and elastic property calculations. The Projector Augmented Wave (PAW) was used to characterise electron-ion interactions[13]..For the exchange-correlation functional, the spin-polarised generalised gradient approximation parameterised by Perdew, Burke and Ernzerhof (GGA-PBE) [14] was chosen. A total energy cut-off of 400 eV was adopted since it was adequate to converge the total energy of the tetragonal NdREFe14B permanent magnets (RE = Gd, Ce). As per Monkhorst and Pack [15], an appropriate k-point mesh of 6x6x3 was used to converge the total energy of the structures. Before calculating any properties, geometry optimisation was done to fully relax each structure. With a strain of 0.005, the calculated elastic constants for every structure were obtained. Following that, the Poisson ratio, bulk modulus, shear modulus, and elastic modulus were computed.

Results and Discussions

Structural and Thermodynamic properties

Table 1 shows the calculated lattice parameters for Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets. The calculated a parameter for Nd2Fe14B is 8.25 Å, which is comparable with the experimental value of 8.82 Å [7]. Moreover, the c parameter was found to be 12.12 Å, and this is 1% comparable with the experimental value (12.25 Å Ref). Nd was substituted with both Gd and Ce rare earth elements. NdCeFe14B magnets have lattice parameters of a = 8.63 Å and c = 12.95 Å. We note that for Nd substitution with Gd, the a parameter increased from 8.25 Å to 8.29 Å whereas the c parameter decreased from 12.12 Å to 11.47 Å. The decrease in the c parameters is due to the small atomic radius of Gd (1.80 Å) as compared to Nd (2.06 Å).

Table 1: Lattice parameters of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets.

Structures	Parameters	Calculated (Å)	Experimental (Å)
Nd2Fe14B	a	8.25	8.82
c	12.12	12.25
NdGdFe14B	a	8.29
c	11.47
NdCeFe14B	a	8.63
c	11.95

The heats of formation (∆Hf) for Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets are calculated as shown in Table 2. ∆Hf are calculated to evaluate the thermodynamic stability of alloys, where negative ∆Hf display a stable material, whereas the positive ∆Hf alludes to the instability of the material. The following equation estimates the [16];

(1)

where is the calculated total energy of the compound, is the calculated total energy of element i in the compound and xi is the composition of each element. The computed heats of formation for Nd2Fe14B was found to be -6.49 eV/atom, this indicates that the magnet is thermodynamically stable. Substitution of Nd with Gd and Ce displays a slight decrease in the calculated ∆Hf. NdGdFe14B has a calculated ∆Hf of -6.26 eV/atom, which is less than that of Nd2Fe14B magnets. Furthermore, NdCeFe14B permanent magnets have the ∆Hf with the value of -6.35 eV/atom. These results suggest that both NdGdFe14B and NdCeFe14B permanent magnets are thermodynamically stable since ∆Hf was found to be negative.

Table 2: Heats of formation of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets.

Structures	∆Hf (eV/atom)
Nd2Fe14B	-6.49
NdGdFe14B	-6.26
NdCeFe14B	-6.35

Elastic properties

Table 3 shows the elastic constants for Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets. Elastic constants are calculated to determine the mechanical stability of the magnets, response to externally induced stress and interatomic bonding conditions. The material is considered to be mechanically stable if it satisfies the stability criterion set for that crystal structure. The following is the condition of tetragonal crystal stability [17]:

(2)

Calculated elastic constants for Nd2Fe14B satisfy the Born stability criterion, suggesting mechanical stability. Moreover, NdGdFe14B and NdCeFe14B permanent magnets are mechanically stable with the computed elastic constants satisfying the Born stability criterion. The stability of the magnets is maintained by replacing Nd with Gd and Ce.

Table 3: Elastic constants of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets.

Structures	Nd2Fe14B	NdGdFe14B	NdCeFe14B
C11	298	382	215
C12	159	143	51
C13	66	141	23
C33	94	333	160
C44	66	113	24
C66	120	117	117
C’	69	120	83

Pugh’s ratio, Poisson’s ratio and Cauchy pressure are calculated to determine the ductility of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets as shown in Table 4. Pugh’s ratio (k = B/G) is used to assess the ductility of these magnets. The material is considered ductile when k is greater than 1.75; else, the material is brittle as proposed by Pugh [18]. Nd2Fe14B magnets were found to be brittle since the calculated k with a value of 1.13 is less than the critical value of 1.75. Substitution of Nd with other REs such as Ce and Gd is considered to enhance the ductility of Nd2Fe14B magnets. Gd substitution displays an increase in the value of k to 1.91, which alludes to the ductility of NdGdFe14B magnets. Moreover, the substitution of Nd with Ce shows an increase in the value of k to 1.58. NdCeFe14B permanent magnets are brittle since k is less than 1.75. Gd substitution enhances the ductility of Nd2Fe14B permanent magnets.

Table 4 also shows the Poisson’s ratio (v) of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets. Material is considered to be ductile when the v is greater than 0.26; otherwise, it is brittle. Nd2Fe14B is observed to be brittle since the calculated v is less than 0.26 with a value of 0.20. The calculated v is in good accordance with the experimental value within 5% [19]. From our results, we notice that v is greater than 0.26 for NdGdFe14B, indicating ductility. Ce substitution has enhanced the Poisson’s ratio of Nd2Fe14B to 0.24, and NdCeFe14B is brittle. For the Cauchy pressure, a positive value indicates ductility, otherwise brittle[20]. Similarly to Pugh’s and Poisson’s ratios, both Nd2Fe14B and NdCeFe14B are brittle while NdGdFe14B is ductile. Ductility of Nd2Fe14B is enhanced by substituting Nd with Gd.

Table 4: Pugh’s ratio, Poisson’s ratio and Cauchy pressure for Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets.

Structures	Nd2Fe14B	NdGdFe14B	NdCeFe14B
B/G	1.13	1.91	1.58
v	0.20	0.28	0.24
Cauchy Pressure	-0.18	29.52	-1.28

Magnetic properties

Table 5 presents the calculated total spin magnetic moments for Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets. We note that our findings on the magnetic moments are in good accordance with previous calculations [21]. The total spin magnetic moments for Nd2Fe14B is 35.63 µB. Substitution of Nd with Ce and Gd compromises the total spin magnetic moments of Nd2Fe14B. The calculated magnetic moments of the NdCeFe14B and NdGdFe14B magnets were found to be 26.05 µB and 27.95 μB, respectively, which is less than that of the Nd2Fe14B magnets. This suggests that magnetic moments of the Nd2Fe14B magnet were reduced due to Ce and Gd having less number of unpaired electrons. Gd was reported to be beneficial in enhancing the Curie temperature of Nd2Fe14B and other magnetic properties [10]; however, it reduces the magnetic moments of the alloy.

Table 5: Magnetic moments of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets.

Structures	Nd2Fe14B	Exp.	NdGdFe14B	NdCeFe14B
Magnetic Moments	35.63	35.00	27.95	26.05

Conclusions

The first-principle approach was employed to investigate the structural, thermodynamic, mechanical and magnetic properties of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets. The calculated structural properties are in good accordance with experimental data within 5%. Energy of formation predicts the thermodynamic stability of Nd2Fe14B, NdGdFe14B and NdCeFe14B permanent magnets. Mechanical properties suggest that Gd substitution enhanced the ductility of Nd2Fe14B magnets. We found that Nd₂Fe₁₄B exhibits a high magnetic spin value of 35.63 µB, followed by NdGdFe₁₄B. Analysis of magnetic and mechanical properties predicted NdGdFe14B alloys to be promising candidates for magnetic applications such as electric vehicles and wind turbines.

References

[1]	Q. Gong, M. Yi, R. F. L. Evans, O. Gutfleisch, and B. X. Xu, “Anisotropic exchange in Nd–Fe–B permanent magnets,” Mater Res Lett, vol. 8, no. 3, pp. 89–96, 2020, doi: 10.1080/21663831.2019.1702116.

[2]	M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and Y. Matsuura, “New material for permanent magnets on a base of Nd and Fe (invited),” J Appl Phys, vol. 55, no. 6, pp. 2083–2087, 1984, doi: 10.1063/1.333572.

[3]	X. Yin et al., “An Efficient Process for Recycling Nd–Fe–B Sludge as High-Performance Sintered Magnets,” Engineering, vol. 6, no. 2, pp. 165–172, Feb. 2020, doi: 10.1016/j.eng.2019.11.007.

[4]	J. Zheng et al., “Magnetic Properties and Microstructure of Ce-Cu-Al Low Melting Alloy Bonding Sm2Fe17N3 Magnet Fabricated by the Hot-Pressing Method,” Magnetochemistry, vol. 8, no. 11, pp. 1–16, 2022, doi: 10.3390/magnetochemistry8110149.

[5]	L. Li et al., “Micromagnetic simulations on demagnetization processes in anisotropic Nd 2 Fe 14 B magnets,” Journal of Rare Earths, vol. 37, no. 6, pp. 628–632, Jun. 2019, doi: 10.1016/j.jre.2018.10.018.

[6]	J. Yu, S. Jiang, D. Sun, P. Lin, and Y. Zhang, “Correlation of mechanical properties and electronic structures for NdFeB permanent magnet under hydrostatic pressure based on first-principle calculation,” Journal of Materials Research and Technology, vol. 18, pp. 3410–3427, May 2022, doi: 10.1016/j.jmrt.2022.03.189.

[7]	M. Sagawa, K. Hiraga, H. Yamamoto, and Y. Matsuura, “Permanent magnet materials based on the rare earth-iron-boron tetragonal compounds (invited),” IEEE Trans Magn, vol. 20, no. 5, pp. 1584–1589, 1984, doi: 10.1109/TMAG.1984.1063214.

[8]	M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura, and S. Hirosawa, “Magnetic properties of rare‐earth‐iron‐boron permanent magnet materials,” J Appl Phys, vol. 57, no. 8, pp. 4094–4096, Apr. 1985, doi: 10.1063/1.334629.

[9]	M. E. Baloyi, R. G. Diale, P. E. Ngoepe, and H. R. Chauke, “Effect of Terbium on the magnetic properties of Nd2Fe14B permanent magnets,” MRS Adv, vol. 10, pp. 1123–1126, 2025, doi: 10.1557/s43580-025-01295-8.

[10]	J. Zheng et al., “Enhanced thermal stability of nanocrystalline melt-spun Nd-Pr-Fe-B alloys by Gd substitution,” J Magn Magn Mater, vol. 604, p. 172316, 2024, doi: https://doi.org/10.1016/j.jmmm.2024.172316.

[11]	G. Kresse and J. Hafner, “Ab. initio molecular dynamics for liquid metals,” Phys Rev B, vol. 47, p. 222, 1994.

[12]	W. Kohn and L. J. Sham, “Self-Consistent Equations Including Exchange and Correlation Effects,” Phys Rev B, vol. 140, pp. 1133–1138, 1965.

[13]	G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys Rev B, vol. 59, pp. 1758–1175, 1999.

[14]	J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized Gradient Approximation Made Simple,” Phys Rev Lett, vol. 77, pp. 3865–3868, 1996.

[15]	H. J. Monkhorst and J. D. Pack, “Special points for Brillonin-zone integrations*,” Phys Rev B, vol. 13, pp. 5188–5192, 1976.

[16]	R. E. Watson and L. H. Bennett, “Heats of Formation of Transition-Metal Alloys,” Phys Rev Lett, vol. 43, pp. 1130–1132, 1979.

[17]	M. J. Phasha, P. E. Ngoepe, H. R. Chauke, D. G. Pettifor, and D. Nguyen-Mann, “Link between structural and mechanical stability of fee- And bcc-based ordered Mg-Li alloys,” Intermetallics (Barking), vol. 18, no. 11, pp. 2083–2089, 2010, doi: 10.1016/j.intermet.2010.06.015.

[18]	J. W. Glen and S. F. Pugh, “The effect of phase transformations on the orientation of zirconium crystals,” Acta Metallurgica, vol. 2, no. 3, pp. 520–529, 1954, doi: https://doi.org/10.1016/0001-6160(54)90073-5.

[19]	G. Tian et al., “Ab initio calculation of electronic structure and magnetic properties of R2Fe14BNx (R = Pr,Nd),” AIP Adv, vol. 8, no. 5, p. 1, 2018, doi: 10.1063/1.5007022.

[20]	D. G. Pettifor, “Theoretical predictions of structure and related properties of intermetallics,” Materials Science and Technology, vol. 8, no. 4, pp. 345–349, 1992, doi: 10.1179/mst.1992.8.4.345.

[21]	A. E. Aly, “Density of States of Rare-earth Permanent Magnet Nd 2 Fe 14 B Using Spin-orbit Coupling,” International Journal of Pure and Applied Physics, vol. 5, no. 3, pp. 215–229, 2009, [Online]. Available: http://www.ripublication.com/ijpap.htm