First-principles study of electronic structure and magnetic properties of L10-ordered FeNi, FePd, and FePt alloys.

The structural, electronic, and magnetic properties of three spin configurations of L10-ordered FeM alloys (M = Ni, Pd, or Pt) were studied using the first-principles method. The calculations were carried out using Quantum ESPRESSO package within the framework of Density Functional Theory (DFT). The exchange-correlation functional potentials were studied using local density approximation (LDA) of Perdew-Zunger (PZ), the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE), Perdew and Wang 91 (PW91), and Perdew-Burke-Ernzerhof revised for solids (PBEsol). We found that the PBE approximation has the most accurate results for lattice parameters compared to the experimental values. Furthermore, our results reveal that the most stable spin configuration for the considered alloys is the ferromagnetic configuration, where all spins are aligned perpendicular to the (001) plane. However, in FePd and FePt alloys, a small variation in the tetragonality ratio c/a (from 0.98 to 0.92) can transform them from ferromagnetic to antiferromagnetic state. In an antiferromagnetic state, a pseudogap is observed just below fermi energy for each alloy. Moreover, our calculations reveal large magnetocrystalline anisotropies for FePt alloy in the order of 3 meV/f.u. On the other hand, FePd and FeNi show relatively lower values in the range of 0.18-0.42 meV/f.u. Finally, Heisenberg exchange interactions are calculated from first-principles and Green's functions formalism.

Introduction

In the last decade, transition metal alloys were the focus of numerous studies in the last decade due to their potential applications in many fields. These applications include magneto-optical recording technologies [1, 2, 3], magnetic media [4], magnetic sensors [5, 6], and catalytic applications [7, 8, 9]. The main attraction forms of these alloys are nanoparticles and thin films [10, 11]. The key factor for these alloys is their divers’ magnetic properties and the ability to control their parameters by adjusting their growth formula and conditions. For instance, if these alloys are engineered to have large saturation magnetization (Ms), high Curie temperature (TC), and large magnetocrystalline anisotropy energy (MCA), then they can be employed in many technological applications such as electric motors and storage devices [12].

Magnetocrystalline anisotropy energy is the difference in the total energy for spin moment alignment along different crystallographic axes, typically termed an easy and a hard axis. It originates from spin-orbit interactions, which are considered short-range interactions [13, 14, 15]. For example, it has been shown that the MCA for FePt alloy can be substantially increased up to the order of 10 MJ/m3 [16, 17, 18, 19, 20] when it is in the L10 ordered phase. Similarly, FePd [21, 22] and FeNi [14, 23, 24] alloys in the L10 ordered phase showed relatively high MCA values.

According to work done by Brown et al. on FePt alloys in the L10 phase. [25], strong competition is present between ferromagnetism (FM) and antiferromagnetism (AFM) states. Their results reveal that the FM state is more favorable as tetragonal distortion decreases. Shuttleworth [26] employed spin-polarized density functional theory (DFT) to study the band structure of binary PtxM1−x (M = Fe, Co, and Ni) alloys in the ordered L12 and L10 phases.

Werwiński et al. [27] calculated MCA for L10 FeNi using generalized gradient approximation (GGA) with WIEN2K, FLPO, and FP-SPR-KKR code. The estimated value is less than 0.5 MJ m−3 which is below the reported values in the literature. This expected limitation in the GGA result can be improved by including additional factors such as tetragonal strain and alloying [27]. Burzo et al. [28] studied band structure and magnetic properties of FePd ordered compounds using first-principles calculations. Their calculations for magnetic moments are in very good agreement with the reported experimental values.

In this work, structural, electronic, and magnetic properties of L10-ordered binary FeNi, FePd, and FePt alloys are investigated using several exchange-correlation approximations. This work includes calculating Lattice parameters, bulk modulus, the total and partial density of states, total and partial magnetic moment, magnetocrystalline anisotropy, Hubbard parameter, and Heisenberg exchanges interactions. The achieved results are discussed and compared to the available experimental and theoretical values.

Computational method

Fully L10-ordered FeNi, FePd, and FePt alloys were considered in three collinear spin configurations: antiferromagnetic (AFM), ferrimagnetic (FIM), and ferromagnetic (FM), as shown in Figure 1, Figure 2. Quantum ESPRESSO [29, 30] was used for spin density functional calculations. The Projector Augmented Wave method (PAW) [31, 32, 33] was utilized to describe electron-ion interactions where the valence electron configurations for iron, nickel, palladium, and Platinum are, respectively, as follows: 3d64s2, 3d84s2, 4d10, 5d96s1. The exchange-correlation functional potentials were studied using: local density approximation (LDA) of Perdew-Zunger (PZ) [34], the generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) [35, 36], Perdew and Wang 91 (PW91) [36], and Perdew-Burke-Ernzerhof revised for solids (PBEsol) [37].

Figure 1

(a) L10 structure of FeM (M = Ni, Pd, or Pt) consist of two possible unit cells (b) Schematic diagram of FCT unit cell (c) Schematic diagram of BCT unit cell.

Figure 2

Antiferromagnetic (Model 1), ferrimagnetic (Model 2), and ferromagnetic (Model 3) configurations of fully L10-ordered FeM (M = Ni, Pd, or Pt) alloys.

A wave-function kinetic energy cutoff of 75 Ry, a charge density cutoff of 500 Ry, and a first-order Methfessel–Paxton smearing of 0.02 Ry were used throughout this work. In addition, a low value of mixing parameter (0.05) was used to ensure convergence of the self-consistent field to the lowest Kohn–Sham energy state.

The equilibrium lattice parameters a presented in Table 1 for L10 structures were obtained by letting the lattice to relax along the c direction for each value of a. Then, the equation of state for each alloy was calculated and numerically minimized. Finally, the equation of state was obtained by plotting the density functional theory (DFT) Kohn–Sham energy as a function of volume. The achieved equilibrium lattice parameters with this approach were used throughout the remainder of this work.

Table 1

The lattice parameters (a and c) and the bulk modulus for FeM (M = Ni, Pd, or Pt) calculated for the configurations illustrated in Figure 2.

Cal. Method	a (Å)				c (Å)				B (GPa)	dB
	PZ	PBE	PBEsol	PW91	PZ	PBE	PBEsol	PW91	PBE	PBE
FeNi
Model 1	3.41	3.55	3.49	3.55	3.41	3.55	3.49	3.55	164.7	5.96
Model 2	3.43	3.57	3.48	3.54	3.43	3.57	3.47	3.59	172.4	5.57
Model 3	3.47	3.57	3.49	3.55	3.47	3.57	3.56	3.62	182.5	5.43
Theor.	3.56 [27]				3.58 [27]				194
Exp.	3.57 [44]				3.57 [44]				165–177	-
FePd
Model 1	3.78	3.94	3.86	3.92	3.63	3.63	3.55	3.60	145.4	4.81
Model 2	3.78	3.86	3.83	3.87	3.65	3.78	3.75	3.78	151.4	5.29
Model 3	3.76	3.84	3.78	3.84	3.68	3.76	3.71	3.76	159.2	5.04
Theor	3.80 [28]				3.66 [28]				-	-
Exp.	3.855 [45]				3.714 [45]				-	-
FePt
Model 1	3.81	3.92	3.89	3.92	3.58	3.60	3.58	3.60	180	5.02
Model 2	3.83	3.86	3.92	3.87	3.62	3.76	3.65	3.59	186.7	5.21
Model 3	3.76	3.86	3.81	3.86	3.68	3.79	3.73	3.79	194.8	5.07
Theor	3.89 [26]				3.705 [26]				-	-
Exp.	3.85 [46], 3.86 [47]				3.713 [46], 3.79 [47]				-	-

For MCA calculation, the spin-orbit coupling included in Kohn–Sham equation was employed in the force theorem to get more accurate results. Two sets of self-consistent field calculations were performed for each alignment of the magnetic moment. One oriented parallel to the easy axis (-axis) yields the energy EII and the other aligned perpendicular to the easy axis yields the energy E⊥. Thus, the MCA can be given by the following equation.MCA = E

Furthermore, The Dudarev [38] approach was applied where we considered only Hubbard effective parameter U = U – J, here U and J are Coulomb and exchange parameters, respectively. J is considered zero. Hence, U is equal to U in our calculations. The values of this parameter U presented in Table 2 were calculated for the elements used in this study using a technique proposed by Cococcioni [39]. Thus, enabling us to study the effect of onsite Coulomb interaction (U) under (GGA + U) and (LSDA + U) formulation of the DFT.

Table 2

The total energy difference between the three considered models and the calculated Hubbard effective parameters for each atomic type.

Alloy	Model	ΔE (mRy)	UFe (eV)	UM (eV)
FeNi	Model 1	46.97	1.404	2.783
	Model 2	5.37
	Model 3	0.0
FePd	Model 1	59.48	1.905	0.062
	Model 2	3.82
	Model 3	0.0
FePt	Model 1	62.58	1.671	0.147
	Model 2	9.34
	Model 3	0.0

The Heisenberg exchange parameters were calculated using Green's functions formalism by adopting the method proposed by Korotin et al. [40]. In this case, a supercell of 16 atoms consisting of 2 × 2 × 2 unit cells is considered, as shown in Figure 1b. The purpose of using supercell is to include all possible exchange interactions. Based on this structure, five parameters are present: JFe1, JFe2, JFeM, JM1, and JM2, where M indicates Ni, Pd, or Pt. These parameters are shown in Figure 3.

Figure 3

Schematic view of FeM (M = Ni, Pd, or Pt) supercell. Five types of Heisenberg exchange interactions between Fe–Fe, Fe-M, and M-M atoms are shown.

Results

Crystal structures & bulk modulus

The FeM (M = Ni, Pd, or Pt) alloys considered in this study are crystallized in a tetragonal L10 structure with space group No. 123 (P4/mmm) with two possible unit cells shown in Figure 1. The first unit cell is FCT (shown in Figure 1b) consist of four atoms: two Fe atoms occupy (0, 0, 0) and (½, ½, 0) positions, and two M atoms occupy (½, 0, ½) and (0, ½, ½) positions. The second unit cell is BCT (shown in Figure 1c) contains two atoms: Fe atom occupies (0, 0, 0) lattice site and M atom occupies (½, ½, ½) lattice site.

The lattice parameters for the FCT unit cell were calculated using different approximation methods for the three models shown in Figure 2. These models are antiferromagnetic (Model 1), ferrimagnetic (Model 2), and ferromagnetic (Model 3). The results for our calculations are shown in Table 1, alongside the corresponding experimental and theoretical values from the literature for comparison. The total energy differences between the considered models are shown in Table 2. Based on our results, the first-principles calculations for a complete lattice relaxation reveal that model 3 is the most stable configuration. This conclusion coincides with the experimental outcome from the literature [41]. Furthermore, the PBE approximation results are the closest to the cited experimental values. Thus, the lattice parameters of PBE will be used from hereafter.

For bulk modulus calculations, the total energy is calculated using PBE approximation at several points around the equilibrium volume. Then, these data will be fitted using the third-order Birch–Murnaghan equation of state [42], as shown in Figure 4. From the fitting data, the bulk modulus B0 and its derivative for the ordered L10 FeNi, FePd, and FePt can be estimated as shown in Table 1. For FeNi as an example, our calculations yield B0 = 182.5 GPa at 0K. This is very close to the range of the experimental values for Fe0.5Ni0.5 alloy at room temperature, which is expanding from 165 GPa to 177 GPa [42, 43].

Figure 4

The total energy per unit cell as a function of the volume for a) FeNi, b) FePd, and c) FePt. The dots are the data points obtained from the PBE approximation. Solid line: Third-order Birch–Murnaghan equation of state fitting.

Density of states

The tetrahedron method is used to calculate the total density of states (TDOS) and the partial density of states (PDOS) for the three configuration models: antiferromagnetic (Model 1), ferrimagnetic (Model 2), and ferromagnetic (Model 3), as illustrated in Figure 2. Our results are shown in Figure 5, where the spin-up density of states (DOS) is indicated by positive values, and negative values indicate spin-down (DOS) while the Fermi level is set to zero.

Figure 5

Total density of states (top panel) and projected density of states (bottom three panels) for each model of FeNi, FePd, and FePt. EF is the Fermi energy.

It is clear from Figure 5 (top three panels) that models 2 and 3 have very strong magnetic behavior due to the asymmetry between spin-up and spin-down TDOS. On the contrary, the symmetry of the TDOS in model 1 indicates a zero-sum of the total magnetic moment. Fermi energy slightly fluctuates among the studied alloys due to the small difference between the number of valence electrons for Ni, Pt, and Pd. For both FePd and FePt in Model 1 (AFM), an obvious deep minimum in the TDOS is present. Similarly, this minimum or “pseudogap” also occurs in the TDOS for FeNi in model 1 but is much weaker. For each alloy, this minimum is below Fermi energy. For models 2 and 3, the d-states (DOS) for Fe and M atoms (M = Ni, Pd, or Pt) noticeably overlap with each other. This occurs just below fermi energy in the majority spin states, indicating a strong hybridization between them.

Magnetic moment & magnetocrystalline anisotropy

The calculations of magnetocrystalline anisotropy (MCA) and Heisenberg exchange interaction parameters () are very sensitive to the number of k-points. Thus, it is very important to choose enough k-points to sample the reciprocal space accurately. Figure 6 shows the convergence test for both MCA and for model 3 of the three different alloys used in this work. Based on the analysis of these data, the number of k-points in the first Brillouin zone needed for accurately estimating MCA and are 5832 and 5184, respectively. The difference between the number of k-points for MCA and is due to the difference in the c/a ratio. For MCA, we used an FCT structure with c/a ∼0.95 corresponding to a sampling of 18 × 18 × 18 k-points, while for a BCT structure is used with c/a∼ 1.35 corresponding to a sampling of 18∗18∗16 k-points.

Figure 6

Convergence test for both magnetocrystalline anisotropy (MCA) (left panels) and Heisenberg exchange interaction (J) (right panels) for model 3 (FM) for a) FeNi, b) FePd, and c) FePt. Eijk indicates the direction of the magnetic moments (including spin-orbit coupling).

To study the magnetocrystalline anisotropy for the three examined models, we considered the alignment of the magnetic moments along the directions [100], [110], and [001]. Our results are presented in Table 3 alongside the corresponding experimental data from the literature Focusing on model 3, which has the most stable configuration, we found that: MCA for FeNi is within the range of the reported experimental values, MCA for FePt is slightly above, and MCA for FePd is clearly beyond the reported experimental values.

Table 3

Total magnetic moments, the magnetic moment per atom, and magnetocrystalline anisotropy (MCA) corresponding to three possible directions ([100], [110], and [001]) for each model of the three considered models shown in Figure 2.

		MFe (μB/atom)	MM (μB/atom)	M (μB/f.u.)	MCA GGA (meV/f.u.)	MCA GGA + U
						(meV/f.u.)	(MJ/m3)
FeNi
Model 1	001	2.34	0.0	0.0	0.015	0.128	0.89
	100	2.34	0.0
	110	2.34	0.0
Model 2	001	2.68	0.67	6.52	0.023	0.151	1.06
	100	2.68	0.68
	110	2.67	0.68
Model 3	001	2.67	0.68	6.52	0.053	0.181	1.27
	100	2.67	0.68
	110	2.67	0.68
Theor.		2.74 [53]	0.70 [53]		-	0.067 [53] 0.11 [54] 0.11 [12]	0.47 [53]0.78 [54]0.77 [12]
Exp.		2.6 ± 0.16 [55]	0.73 ± 0.0.4 [55]		-	-	0.58–1.3 [54,56]
FePd
Model 1	001	2.88	0.0	0.0	0.092	0.241	1.69
	100	2.88	0.0
	110	2.88	0.0
Model 2	001	3.03	0.38	6.67	0.028	0.171	1.20
	100	3.03	0.38
	110	3.02	0.38
Model 3	001	3.02	0.38	6.65	0.016	0.417	2.92
	100	3.03	0.38
	110	3.02	0.37
Theor.		2.81 [57]	0.40 [57]	6.42 [57]	-	0.29 [58]	1.70 [58]
Exp.		3.01 [59]	0.30 [59]	6.62 [59]	-	-	1.5 [21], 2.1 [22]
FePt
Model 1	001	2.88	0.0	0.0	0.41	1.62	11.33
	100	2.88	0.0
	110	2.89	0.0
Model 2	001	3.02	0.38	6.66	0.26	1.42	9.93
	100	3.02	0.37
	110	3.02	0.38
Model 3	001	3.02	0.38	6.67	1.73	3.203	22.42
	100	3.02	0.38
	110	3.01	0.38
Theor.		2.92 [60]3.11 [61]	0.33 [60] 0.39 [61]	-	-	2.68 [61] 2.90 [60]	-
Exp.		-	-	-	-	1.9–3.16 [16,62,63]	-

Experimentally, due to the difficulty synthesizing perfectly ordered samples and the complexity in cooling the sample close to the absolute zero, the measured value of (MCA) tends to be always lower than the theoretical value [14, 48]., which is the case here. In addition, the orbital polarization correction was not considered in our calculations. This factor has been reported to increase the MCA even more [49, 50].

FePt alloy showed a significant magnetocrystalline anisotropy due to the large spin-orbit coupling of heavy Pt atom and a strong hybridization of the d bands of Pt and Fe [51], making spin-orbit interactions much stronger. On the other hand, for the FePd and FeNi alloys, the contribution of the relatively lighter atoms Pd and Ni is very small since the magnetocrystalline anisotropy mainly originates from orbital magnetic moments of Fe through spin-orbit interaction [52].

Furthermore, the atomic magnetic moment differences along the considered directions are very small (<0.01 μB/atom). Consequently, it is hard to specify the most energetically favorable direction for the alignment of the magnetic moment within the accuracy limit of our calculations.

Heisenberg exchange interaction

Finally, Table 4 shows the Heisenberg exchange interaction parameters ( for all models. Similar atoms display pronounced positive exchange interactions, indicating that Fe spins in each plane would prefer to align ferromagnetically. On the other hand, different atoms in model 2 show negative interaction parameters indicating that different atoms have opposite spin signs. The exchange interactions between atoms along the -axis will be identical to the interactions along the -axis since they share the same distances. The importance of calculating Heisenberg exchange interaction parameters is to calculate Curie temperature for these structures. Curie temperature can be calculated using many approaches, and the most popular one is using mean-field theory (MFT). However, this method suffers from a lack of accuracy. The alternate method is implementing Monte Carlo (MC) simulation, which is well known for its accuracy.

Table 4

Heisenberg exchange interaction parameters (meV) for model 1 (Antiferromagnetic), model 2 (Ferrimagnetic), and model 3 (Ferromagnetic).

		JFe1	JFe2	JM1	JM2	JFeM
FeNi	Model 1 (AFM)	0.849	-8.157	0.216	2.671	5.581
	Model 2 (FIM)	0.573	7.291	0.161	3.223	-5.352
	Model 3 (FM)	1.419	8.631	0.261	3.416	5.785
FePd	Model 1 (AFM)	1.927	-15.478	0.241	0.492	5.918
	Model 2 (FIM)	2.052	14.359	0.088	0.553	-5.431
	Model 3 (FM)	2.551	14.352	0.302	0.504	5.401
FePt	Model 1 (AFM)	1.824	-13.734	0.011	0.581	5.475
	Model 2 (FIM)	4.056	12.642	0.999	0.296	-6.638
	Model 3 (FM)	0.903	11.101	0.193	0.602	5.304

Conclusion

We have performed the first principals’ calculations on L10-ordered FeNi, FePd, and FePt alloys in three different spin configurations. The structural, electronic, and magnetic properties were investigated. Structural properties such as lattice parameters, bulk modulus, and its derivative were computed by implementing several approximations for the exchange-correlation functional potentials. In this regard, the results of the PBE approximation proven to be the closest to the experimental values. We showed that ferromagnetic configuration is the most stable configuration for the considered alloys. For ferrimagnetic and ferromagnetic states, a strong hybridization between d orbits of the compound atoms is found below fermi energy in the majority spin states. For the antiferromagnetic state, a pseudogap is observed below fermi energy. Magnetocrystalline anisotropy (MCA) is calculated for each alloy in all spin configurations. FePt alloy showed an exceptionally large MCA value in the order of 3 meV/f.u. Finally, we calculated Heisenberg exchange interaction parameters for ferrimagnetic and ferromagnetic states. These parameters can be utilized for further studies using Monte-Carlo simulations.

Declarations

Author contribution statement

K. Aledealat: Analyzed and interpreted the data; Wrote the paper.

B. Aladerah: Performed the experiments.

A. Obeidat: Conceived and designed the experiments.

M. Gharaibeh: Analyzed and interpreted the data.

Funding statement

This research was supported by the Jordan University of Science and Technology (82/2021).

Data availability statement

Data will be made available on request.

Declaration of interests statement

The authors declare no conflict of interest.

Additional information

No additional information is available for this paper.